Under $1000? No.
Sports photography is a very demanding on gear and the photographer. The cheapest body I would recommend if you are serious about getting into sports photography is a Canon 7D MkII. It is designed for sports and is my current third body for sports shooting. The other thing is the need for good, fast glass. That usually means a fixed f2.8 or faster which are also not cheap. The go to first lens for most sports photographer is a 70-200 f2.8. Not cheap but much cheaper than a 300 f2.8 or 400 f2.8.
There is a big difference between a good sports shot and a good sports snap shot. Sell-able shots need to isolate the action which is one of the reasons for fast glass. You need to shoot close to if not wide open to get that small DOF that isolates the action which is why you want fast glass.
With that said you can shoot sports with cheaper consumer grade bodies and lenses however you keeper rate will be lower as will you sell-able rate. Sports photography is now a freelance business and there are a lot of people that like to shoot sports. Gone are the days of Sport Illustrated, Sport Magazine or even large paper staff sports photographer jobs.
I'm not trying to be a downer, everyone has to start somewhere. What ever body you decide on needs to have good high ISO performance and stellar AF performance as well as a decent burst mode. Your glass choice will also be very important. It can be a little less important if you only shoot outdoor sunny weather sports, but those are few and far between these days.
It also helps living in a good location. I shoot college sports these days and while not in a hugely large city I am usually no more than 5 hours drive from many large FBS schools.
To break in, start with high school sports. Shot a lot and practice just as much. Every late summer early fall I shoot the practices for our local colleges just to get back in practice. Build a portfolio. If you can follow a good high school team or teams to various events at small universities etc. Keep in mind everyone loves to shoot football, baseball and basketball, but tennis golf, cross country also are sports and some of them are more likely to take you to the next level where you can make contacts. Always I mean always be professional with every contact you make, even if they turn you down, thank them for the opportunity to talk to them. Today they may turn you down and tomorrow they may be the guy letting you in. Know the sport inside out upside down and sideways. Know the players, their strengths and weaknesses. That means a whole lot of reading in your off time, the sports pages or high school sports magazines if you location or state has such etc. You need to know the sport and the participants so you can anticipate the action. You will not always be right but if you really know the sport you will be right far more often than you are wrong.
For me the highlight of every fall is the OU, OSU game. It is always and action packed contest and is only about 3 hours from where I live. Getting to shoot from the sidelines is a whole different level of enjoyment. Good luck. Hopefully some of the Nikon shooters will chime in on the best Nikon gear for sports. I shoot Canon but that does not mean that there are not other system that just as good or maybe even better. The skill you develop will mean the most but having good tools makes it a whole lot easier.
Definitely peruse this form.
Sports Photography and Photojournalism for Professional Photographers and Photography | SportsShooter.com It will provide you a lot of knowledge of the business today, the gear, etc.
Some other links of interest.
The Perfect Camera Settings for Action and Sports Photography | Photography Mad
Shooting Sports: Tips from the Pros
Working Your Way Up to Become a Successful Sports Photographer