I need to assume a bit - as your initial post is missing some informations.
- both computers are in the same network
- both computers are connected to this local network (be it wired or using wifi) or directly to each other
- computer1 is running linux
To break it down
If both computers are in the same local network, it is possible in general.
They both need access to either the same local router or directly to each other.
In case of direct connection:
Connect the computers using a crossover cable (may not be necessary most modern ethernet cards should auto negotiate the connection)
Then configure the network card on both sides accordingly (i would go for fix ip addresses)
To share the files from computer1 to computer2, you would need to setup some kind of network-share on computer1 (There are several options and protocols here - i.e. ssh ,ftp, samba etc). People often use samba in such cases.
You should be able to find tons of tutorials on how to setup samba, one possible is here
Yes it is possible.