How can we cover a carpet for the long term?
If your soon-to-be-landlord is already making unreasonable demands, I would reconsider your choice to rent with them. Minor carpet staining from traffic is considered to be "wear and tear" in most states, as well as scuffing of the walls and other things that result from simply living in a space.

Unless you're a group of college kids (to which, I could understand creating some fear so that people don't go party-crazy), a "no stains period" policy is unlikely to be legally binding, however, that is a question for a lawyer. Personally, if those are the types of demands the landlord is making, I would go into your lease expecting a bill for every single thing that should be considered "wear and tear". I've often rejected apartments simply because the landlord has alluded to charging for repairs before I have even moved in.

As for a permanent wall-to-wall covering, I can't think of anything permanent that would  would suffer the wear, be water tight, and especially fit a tight budget. If you are set on this rental, I would consider putting your money toward regular professional carpet cleanings and before moving out, and the purchase of some spot-treatment carpet cleaners for when the inevitable spill does occur. 

Finally, if you can't trust your roommates or their guests to share/respect your concerns for the carpet, you're going to be disappointed.

-My 2 cents