It's simple really. Just select the polygon that you want to clip with. Then right-click the vector layer in the table of contents. Convert features to graphics. I generally choose "Draw the converted graphics and draw the features". This will overlay a vector graphic on top of the polygon. Using the arrow select tool, select the graphic.
Once selected, right-click on the DEM (or any raster for that matter), and choose Data > Export Data. In the extent options, choose "Selected Graphics (Clipping). Continue with the other options to your liking, and then save. You should then have a raster clipped to the vector polygon you wanted.
Edit: I should say this isn't the most expedient way to do multiple clips, but if you need a raster clipped to a vector, and you don't have spatial analyst, then it certainly works.