It's very close. A little over a mile; up to a mile and a half, depending on where in Greenwich Village you're going.
Penn Station is in Chelsea, and Chelsea is the neighborhood directly north of the West Village. Greenwich Village is directly east of the West Village, and the East Village is directly east of Greenwich Village.
The distance from Penn Station to Greenwich Village is a little over a mile, mostly south, but slightly southeast.
For future reference, in New York City, 17 to 20 street blocks (north/south blocks) equal a mile, and 5 to 10 avenue blocks (east/west blocks) equal a mile. As you can see, the length of the avenue blocks varies considerably. On the East Side it's more like 8 to 10 avenue blocks to a mile, while on the West Side it's more like 4 to 7 avenue blocks to a mile.
Of course, in Lower Manhattan, these estimations become worthless, since in Lower Manhattan there's no square grid pattern to the streets. They're all tiny and run in zig-zags.
By train, its about 25 minutes
300 miles
It's 5 blocks down from the east side of Penn Station. Penn Station is at 33rd to 34th Streets, between 8th and 7th Avenues.
There are several Penn Stations located in the US but the most famous is by far Pennsylvania Station (known as Penn Station) in New York City, New York. See Related Links for more information about the station.
It's about 7 miles from Penn Station in Chelsea to 1 Police Plaza in lower Manhattan.
29 miles
Hip, trendy bars in the area around Penn Station are few and far between. You would be much better off taking the subway down to the West Village. The West Village is one of the trendiest neighborhoods in the city, and both the A-C-E subway line (the blue line) and the 1 or 2 trains (of the 1-2-3, the red subway line) run directly between Penn Station and the West Village. If you take the A-C-E, then you will be going from Penn Station (34th Street) to West 4th Street. That is TWO STOPS on the A train, and THREE STOPS on the C or E. If you take the 1 train, then you will be going from Penn Station (34th Street) to Christopher Street/Sheridan Square. That is FIVE STOPS. Do NOT take the 2 or the 3 trains, as they are express trains and do not stop at Christopher Street/Sheridan Square. The West Village is absolutely loaded with trendy bars and restaurants. Just walk around a bit and pick one you like.
Penn Station is at 34th Street so it is 20 blocks away from 54th St. You would take the subway two stops north.
About 5 miles
1 min walk. The North Greenwich tube station is on the site of the 02 arena. The station is on the Jubilee line.
It is 20-25 minutes in train 1 to south ferry
It's 8 blocks straight north from Penn Station (34th Street between 8th and 7th Avenues) to the Port Authority Bus Terminal (42nd Street at 8th Avenue).