Determine row size for table

Another way, run this then look at MaximumRecordsize

dbcc showcontig ('YourTableNameHere') with tableresults

As the DBCC SHOWCONTIG is going to be deprecated, Microsoft advises to use the sys.dm_db_index_physical_stats instead. You can get the maximum, minimum and average row size for each index and other stats, too:

SELECT * FROM sys.dm_db_index_physical_stats (DB_ID(N'AdventureWorks2014'), OBJECT_ID(N'[sales].[SalesOrderHeader]'), NULL, NULL , 'DETAILED')

Output:

database_id object_id   index_id    partition_number index_type_desc                                              alloc_unit_type_desc                                         index_depth index_level avg_fragmentation_in_percent fragment_count       avg_fragment_size_in_pages page_count           avg_page_space_used_in_percent record_count         ghost_record_count   version_ghost_record_count min_record_size_in_bytes max_record_size_in_bytes avg_record_size_in_bytes forwarded_record_count compressed_page_count
----------- ----------- ----------- ---------------- ------------------------------------------------------------ ------------------------------------------------------------ ----------- ----------- ---------------------------- -------------------- -------------------------- -------------------- ------------------------------ -------------------- -------------------- -------------------------- ------------------------ ------------------------ ------------------------ ---------------------- ---------------------
6           1266103551  1           1                CLUSTERED INDEX                                              IN_ROW_DATA                                                  3           0           0.439882697947214            28                   24.3571428571429           682                  98.8644304423029               31465                0                    0                          154                      195                      171.487                  NULL                   0
6           1266103551  1           1                CLUSTERED INDEX                                              IN_ROW_DATA                                                  3           1           0                            2                    1                          2                    54.7442550037064               682                  0                    0                          11                       11                       11                       NULL                   0
6           1266103551  1           1                CLUSTERED INDEX                                              IN_ROW_DATA                                                  3           2           0                            1                    1                          1                    0.296515937731653              2                    0                    0                          11                       11                       11                       NULL                   0
6           1266103551  2           1                NONCLUSTERED INDEX                                           IN_ROW_DATA                                                  2           0           0                            4                    25.5                       102                  99.0670990857425               31465                0                    0                          24                       24                       24                       NULL                   0
6           1266103551  2           1                NONCLUSTERED INDEX                                           IN_ROW_DATA                                                  2           1           0                            1                    1                          1                    31.4801087225105               102                  0                    0                          23                       23                       23                       NULL                   0
6           1266103551  3           1                NONCLUSTERED INDEX                                           IN_ROW_DATA                                                  2           0           0                            3                    36.3333333333333           109                  99.8363355572029               31465                0                    0                          26                       26                       26                       NULL                   0
6           1266103551  3           1                NONCLUSTERED INDEX                                           IN_ROW_DATA                                                  2           1           0                            1                    1                          1                    36.3355572028663               109                  0                    0                          25                       25                       25                       NULL                   0
6           1266103551  4           1                NONCLUSTERED INDEX                                           IN_ROW_DATA                                                  2           0           0                            2                    27.5                       55                   98.9285025945145               31465                0                    0                          12                       12                       12                       NULL                   0
6           1266103551  4           1                NONCLUSTERED INDEX                                           IN_ROW_DATA                                                  2           1           0                            1                    1                          1                    11.527057079318                55                   0                    0                          15                       15                       15                       NULL                   0
6           1266103551  5           1                NONCLUSTERED INDEX                                           IN_ROW_DATA                                                  2           0           0                            2                    27.5                       55                   98.9285025945145               31465                0                    0                          12                       12                       12                       NULL                   0
6           1266103551  5           1                NONCLUSTERED INDEX                                           IN_ROW_DATA                                                  2           1           0                            1                    1                          1                    13.5656041512231               55                   0                    0                          18                       18                       18                       NULL                   0

Here is another query I got, but like Tomalak's it is broken because it doesn't take into account things like the overhead for variable length columns.

SELECT OBJECT_NAME (id) tablename
     , COUNT (1)        nr_columns
     , SUM (length)     maxrowlength
FROM   syscolumns
GROUP BY OBJECT_NAME (id)
ORDER BY OBJECT_NAME (id)

Might not the best but this is showing the number of rows + for columns with datatype date,time and so on

SELECT 
    t.NAME AS TableName,
    s.Name AS SchemaName,
    p.rows AS RowCounts,
    SUM(a.total_pages) * 8 AS TotalSpaceKB, 
    CAST(ROUND(((SUM(a.total_pages) * 8) / 1024.00), 2) AS NUMERIC(36, 2)) AS TotalSpaceMB,
    SUM(a.used_pages) * 8 AS UsedSpaceKB, 
    CAST(ROUND(((SUM(a.used_pages) * 8) / 1024.00), 2) AS NUMERIC(36, 2)) AS UsedSpaceMB, 
    (SUM(a.total_pages) - SUM(a.used_pages)) * 8 AS UnusedSpaceKB,
    CAST(ROUND(((SUM(a.total_pages) - SUM(a.used_pages)) * 8) / 1024.00, 2) AS NUMERIC(36, 2)) AS UnusedSpaceMB
FROM 
    sys.tables t
INNER JOIN      
    sys.indexes i ON t.OBJECT_ID = i.object_id
INNER JOIN 
    sys.partitions p ON i.object_id = p.OBJECT_ID AND i.index_id = p.index_id
INNER JOIN 
    sys.allocation_units a ON p.partition_id = a.container_id
LEFT OUTER JOIN 
    sys.schemas s ON t.schema_id = s.schema_id
INNER JOIN  sysobjects so on t.object_id = so.id
INNER JOIN  syscolumns SC on (so.id = sc.id)
INNER JOIN systypes st on (st.type = sc.type)
WHERE 
    t.NAME NOT LIKE 'dt%' 
    AND t.is_ms_shipped = 0
    AND i.OBJECT_ID > 255 
    AND so.type = 'U'
and st.name IN ('DATETIME', 'DATE', 'TIME')
GROUP BY 
    t.Name, s.Name, p.Rows
ORDER BY 
    p.rows DESC

Have a look at INFORMATION_SCHEMA.COLUMNS. You can add the size of each column in a table to give you an idea which tables have the potential to overshoot the 8k/row rule.

SELECT * FROM INFORMATION_SCHEMA.COLUMNS 

You will figure out the rest when you see the query result.