EarthTools has a webservice that covers the US and Europe using the SRTM data. Given a latitude/longitude it returns the height above sea-level in feet and meters. (found from Quakr Viewr) Geonames offers two services using the SRTM data, and also the GTOPO30 from the USGS. Geonames gets bonus points for also returning the results in JSON. Geonames also uses a larger dataset - lands within 60 degrees north and 56 degrees south.
(This is not the only webservice offered through these sites, you can also retrieve countryinfo, timezone, sunrise/sunset etc.)
Update: see note of caution from Donald below.