Absurdly large /proc/kcore file -- what does it mean?

First things first: /proc is a virtual filesystem. Do not back it up. Its contents do not exist on the disk.

From man 5 proc:

/proc/kcore
      This  file  represents  the physical memory of the system and is
      stored in the ELF core file format.  With this pseudo-file,  and
      an unstripped kernel (/usr/src/linux/vmlinux) binary, GDB can be
      used to examine the current state of any kernel data structures.

      The total length of the file is  the  size  of  physical  memory
      (RAM) plus 4KB.

So, I'd be a bit concerned if the kcore is small.

How do you reduce its size? Reduce your RAM.