jamesjiao: 'gym' is short for both 'gymnasium' and 'gymnastics'. Common terms that are related are 'gym class', and 'gym shorts' meaning the gymnastics shorts worn in a gymnastics class which in term is usually held in a (school) gymnasium. The shorter forms for all of these is what you'd normally hear. These are common terms since the 40s in the US, if not earlier. I'm over 70 and these were the terms used when I was in school. No one used the longer terms. Even today I'd refer to a sports center/facility where members swim, play racquetball/squash, do yoga/aerobics, etc. as a 'gym'. The longer term 'gymnastics' is usually only used when referring specifically to the sport, e.g. a gymnastics team in intercollegiate sports or int'l games such as the Olympics. .
macrevel: I think gymnasium should be accepted. Gym is simply an abbreviated form. I think today the sense of 'palestra' is more that of a sports center, a place where people can work out, swim, play basketball, etc. In the States, the term "rec center" is often heard, short for 'recreation center'.
danielgolotta: The sentence needs either the definite article 'the' or the indefinite article 'a' to be the most logical translation. The only situation where "at gym" makes any sense and would be considered proper grammar is when speaking of a gym class as e.g. in school. If you're referring to a sports facility however, omitting an article is absolutely incorrect.