In SWTOR is there somewhere that the chat/messages log is stored locally?

Solution 1:

This question should be revisited, because as of patch 1.2, a combat log can be enabled, which writes combat data to text files on your computer.

First, the logging must be enabled from Preferences > Combat Logging > Enable Combat Logging to File. You do not get to choose a location for the file, instead, the logs are sent to your "My Documents" folder in the "Star Wars - The Old Republic\CombatLogs" subfolder (same place as the Screenshots folder).

Word of warning: the logs are full of much more information than you might be used to seeing in a combat log. For example, here is a single heal on myself:

[12:41:45.387] [@Nathrakh] [@Nathrakh] [Benevolence {812710891618304}] [Event {836045448945472}: AbilityActivate {836045448945479}] ()
[12:41:47.009] [@Nathrakh] [@Nathrakh] [] [Spend {836045448945473}: Force {836045448938502}] (45)
[12:41:47.016] [@Nathrakh] [@Nathrakh] [Benevolence {812710891618304}] [ApplyEffect {836045448945477}: Heal {836045448945500}] (1252)

Not only can you see the final result (a heal for 1252 HP) but the "AbilityActivate" event and the even a "Spend Force" (45) event.
Because of all this extra information, a log parser is recommended. There are several online log parser options, such as: http://swtor.wildking.net/ and http://swtor.askmrrobot.com/combatlog (I am not affiliated with either of these sites).