The ESRI World Elevation Service is a good place to start.
Determine resolution of the data in your AOI
All data used here is "accurate" but obviously depending on your purpose and the size of your Area of Interest you may want a higher or lower resolution DEM to work with. Obviously the higher the resolution the more detailed the terrain is and more descriptive of the actual surface, but with more data takes more processing time.
There are other sources of elevation data that is also free, if you would be willing to describe in more depth your area of interest and what sort of analysis you were wishing to do, it would help point you in the right direction.
Edit: Also you need an ArcGIS Online Subscription to use the World Elevation Service