How to get an elevation profile for a gps track?

I'd like to get a reasonably accurate elevation profile for a track recorded with a GPS (which often has very unreliable altitude data and occasionally none at all, depending on the model.)

Does anyone have any hints on the easiest way to do this. The two techniques I am considering so far are:

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  Using the Google Elevation API

  
  

  This API is relatively easy to use, but still requires a few steps that aren't trivial due to it's usage restrictions: max 512 samples returned per request, and the number of points along the path is limited (by URL length) as well.

  
  

  I expect a gpsbabel simplify filter can be concocted to reduce the track to a suitable number of points (no point in them being closer than 100m or so together due to the resolution of the altitude data), but then the problem remains of how to map this simplified track back onto the original path, since the lengths will differ.

  
  

  Or, if this isn't suitable for automation the best approach may be to let the user select the transect points on a map manually.

  
  


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  Downloading the Shuttle Radar Topography Mission (SRTM) data and doing the query locally.

  
  

  This is something I have no experience with, so any suggestions on how feasible this is are welcome. How big is the data set? What GIS software is required, and can it be scripted in a suitable manner? I'd prefer not to have to write an sampling and interpolation algorithm, that sounds like a pain. What is the likely performance of such an approach? (I need it to be pretty quick and run on a memory-limited VPS webserver...)

  
  

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Some further details to flesh out @MerseyViking's answer re downloading the data from http://srtm.csi.cgiar.org/SELECTION/inputCoord.asp:

There are 72 x 24 tiles, each about a 20mb zip file that decompresses to a 72.1mb 16bit TIF file (6001x6001 pixels).

That's ~120 gb, which is more than I can store. Leaving it compressed and ignoring the oceans will reduce it to maybe 10gb, which is still a bit too large. Loading the data on-demand would dramatically reduce the needed storage space, but the source site is slow (I was only getting 10kb/s) making that pretty impractical.