My answer will mostly be supplemental to /u/sunagainstgold's fantastic linked answer, since they cover many of the important points.

The first point to make is that it's very difficult to generalise the economic circumstances across space and across time in Medieval Europe; what was true for one county in England when it came to serfdom might very well not be true for it's neighbour, never mind trying to compare more disparate areas. So, I'll talk specifically about England around 1150-1300.

Another thing to keep in mind is that being a serf did not necessarily mean that one was the poorest of the poor. A serf might well hold a pretty substantial tenement from his lord, whereas his free neighbour might well be struggling to make ends meet. This is not, of course, to say that serfdom was a desirable state - I think the argument that records of court cases show that the un-free in Medieval England were resentful of their status still holds water.

So this class of peasant whose tenement really was completely inadequate to support them, free or bond, what could they do to make ends meet? They actually formed a fairly important part of the rural economy, in as much as they took whatever work was going, and thus served as an accessible labour pool. An impoverished free holder might find himself helping to tend the land of his wealthier, un-free neighbour. These people were the village carpenter, the smith, the miller's assistant and a whole host of other roles that don't fit neatly into the typical peasant farmer archetype.

So what if serfdom really was the problem for you though? What if what you wanted was to be free of bondage? Legally, you had several options available to you. If you believed that you were being held in bondage unlawfully you could purchase a writ De Libertate Probanda.

The proceedings stemming from this writ could go a number of ways. If you were lucky, the man claiming you as his villein would realise the jig was up and simply not turn up and you would then enjoy free status. If, however, the person claiming you as villein did turn up to court you would then have to produce several blood relatives, and if they are admitted to be free, then you would enjoy free status. If your witnesses were doubted then the men of the neighbourhood would be consulted. Similarly, the man who claimed you as his villein could produce his own witnesses as to your status, who would go through the same process, potentially also requiring the consultation of the men of the neighbourhood.

This was the process either when somebody of free status was reduced to villeinage, or if somebody of apparent villein status suddenly claimed free status.

This was not the only way to become free, however. Their lord could simply manumit the villein, or sell him to another who had the intent to free them (it's earlier and from a different region from the one I'm discussing, but this is worth reading). If this happened the villein was free forever, provable in court by "charter or other lawful means". This freedom was not universal; a villein could be manumitted by one lord, but lawfully claimed as a villein by another.

Now, of course "no person of villein status can seek his freedom with his own money. . . because he could be recalled to villeinage by his lord, because all the chattels of a villein are property of his lord". However, a third party could provide the money for freedom. If this sound like a scenario where a well off villein could give money to a free man to purchase that villein's freedom then, well, yes, it was. Indeed, it is not difficult to imagine a scenario where a villein made a generous gift to their lord who subsequently generously manumitted them.

So, to sum up, serfdom and extreme poverty were not necessarily synonymous. Those peasants whose holdings would leave them in extreme poverty could seek various forms of work and were integral to the rural economy. However, well off villeins who wanted to gain or prove their free status had several options open to them under English common law. Loop holes within that law were also an avenue they could take.

I have a couple of answers that I think might interest you:

• Medieval England: I'm a healthy 18 year old male serf with no wife or kids and a harsh lord of the manor--what's keeping me on the manor instead of walking off in search of better opportunities on another manor or in some town? (Also applies to young women--actually, more frequently it was women who left farms for cities!)
• Were there any other "Peasant Revolts" elsewhere across Europe similar to Wat Tyler's in 14th Century England?

As you can see, the topics are...somewhat different. ;)