When you ask such questions its very helpful to show an image of your data. You say you have nested buffers, in some cases these may be rings (think of donuts) or overlapping polygons. Also what influences your solution is knowing if these nested buffers overlap other sets of nested buffers?
Assuming you have sets of overlapping polygons and no one set overlaps any other then a fairly simply model with an iterator would solve this problem. You essentially step through each buffer and clip out the network then summarize length. So if you have a 1000 buffers you end up with 1000 clipped datasets. If this is done within a sub-model you can collect all the summary tables and fire them off to a merge tool in the master model. If you don't know about embedding models then you need to read the help file.
After the summary tool you may want to add fields and populate with layer names and buffer ID's.