1) As you've written it, the article needs to be "an" rather than "a," because "800," when read aloud, begins with a vowel sound. "He's running an 800-meter race." By the way, since "an" is used solely for euphony, this is not a very serious mistake--it doesn't interfere with understanding and might not even be noticed in conversation.
2) In U.S. English, the sentence "He's running an 800-meter race" is good, correct, natural English, but the meaning is not quite what you intend.
I can explain better with a different example. There is an organized bicycle ride to raise money for charity called "The Pan-Mass Challenge." I might say either
a) "Next weekend, she's riding 202 miles," or
b) "Next weekend, she's riding in a 202-mile bicycle ride for charity" or
c) "Next weekend, she's riding in the Pan-Mass Challenge," or possibly
d) "Next weekend, she's riding the Pan-Mass Challenge."
I would use (a) or (b) when speaking to someone unfamiliar with the ride, (c) or (d) when speaking to someone who knows about it. (a) simply describes how far she's riding. (b) makes it clear that she's not simply riding, but that she is participating in an organized event. The event is described, but not named. (c) or (d) names the event, and assumes that the listener knows what the event is.
These sentences are both correct:
"He's running an 800-meter race" and
"He's running the 800-meter."
In the first, "an 800-meter race" is a description of a race. It's appropriate for a listener who isn't familiar with track and field events. In the second, "the 800-meter" is the _name_ of a standard athletic event.
One more example:
(a) She's walking 732 kilometers this summer.
(b) She's walking on a 732-kilometer pilgrimage this summer.
(c) She's walking the Camino de Santiago this summer.