I’m assuming you’re talking about something feature length?
Network, network, network (to piggyback off of Calix). Here are some ideas on how to expedite the process because $1,000 can go fast because of the one thing everybody needs: food.
Are you budgeting craft services into that $1k? If so, keep your crew and cast to a minimum.
Do you already have all of your equipment? If not, find out what you can borrow and who you can convince (that owns said equipment) that making this movie is a great idea and in their best interest. Renting decent equipment will almost instantly kill the $1k over a week, which isn’t enough time to pull off a feature.
Write a story with the elements you currently have access to, be they actors, locations, props, etc. Tailor the script to them… the faster your actors can ‘get’ the role, the fewer takes you’ll need, the faster the production will go, and the less you’ll have to feed everyone.
Another is to take the concept above and tell a story in a time where this technology is currently widely available, so to excuse a lower-end camera, you basically say that these are real people using a camera like this (a la Paranomal Activity/Blair Witch Project).
What’s the end goal of this $1k project? That will determine if it’s impossible. Also, you might consider making shorts if you haven’t done a feature before. Trust me, it’s a long haul to get a project that big off the ground and the more contacts and experience you have, the better off you’ll be for it.
-Ryan