Or was there no off-time? Was a roman Legionnaire expected to be building, fighting, or resting during his entire tenure?
And if they did have times for game and/or sport, what would they have played?
Dice gambling was an extremely popular Roman pastime. Numerous murals of it being played have been found in Pompey, such as this one: http://www.aerobiologicalengineering.com/wxk116/Roman/BoardGames/rbt2.jpg
And contemprary dice have been found in archaeological sites across the empire: http://www.bibleandscience.com/store/catalog/images/Roman%20dice2.jpg
Dice is a simple game thats pieces (a pair of dice and of course, a cup) are easily transported, and so it was popular in the military for obvious reasons.
As with all responses about Roman history this one cannot be answered wholesale because of the vastness of the Roman Empire, the length of time it endured, and the differences between the 20 to 30 legions that could have been stationed anywhere from Spain to Syria. Each legion had fairly distinct lineage (and different leadership) that would make them have a unique personality.
In Caesar's Legion by Stephen Dando-Collins there is a section about a relatively peaceful time in the 1st century AD where some men of the 10th legion actually sold some of their equipment to include shields, and helmets. They assumed lives as normal citizens of the province and owned shops and raised families - which was very much against their contract.
So to answer part of your question, these men would have had quite a bit of off-time for any activities the average citizen would have had access to in the eastern provinces at the time. It all depends upon the leadership, and whether or not the legion is taking part in a campaign.
