I've seen a lot of wrong answers, and a lot of right (but incomplete) answers. So i'll try to give a complete answer this time

1) It IS possible to create a transaction with 0 fee... But the developers of a lot of modern wallets edited the functions of their programs so 0 fee transactions can not be created... If you want to create a 0 fee transaction, you can do it using the createrawtransaction command of bitcoin core (either from the cli or from the debug window), you can also use the electrum console... There might be other wallets that still allow 0 fee transactions to be generated, but bitcoin core and electrum are the wallets i've actually used for creating 0 fee transactions. Do be carefull here: there is a difference between CREATING a 0 fee transaction (which is easy) and getting the 0 fee transaction BROADCASTED to the network (which is hard).

2) the first main problem AFTER you created a 0 fee transaction is getting it relayed... A transaction will ONLY have a chance of ending up in a block IF it ends up in the mempool of the node of a miner first. A lot of nodes have a minimum relay fee, but if i'm not mistaking, a node owner CAN still modify the minimum relay fee and set a custom parameter so there is no minimum relay fee... The problem is that most nodes do have a minimum relay fee, so won't relay your transaction, so the odds of a 0 fee transaction ending up in the mempool of a mining node is small

3) the second main problem is that a miner has no incentive to add a 0 fee transaction. Historically high priority 0 fee transactions were added to blocks by the miners, but i don't think nowadays miners still honour high priority transactions