You can use either git ls-tree -r -l <revision> <path> to get the blob size at given revision, e.g.
$ git ls-tree -r -l v1.6.0 gitweb/README 100644 blob 825162a0b6dce8c354de67a30abfbad94d29fdde 16067 gitweb/README
From the docs:
-r recurse into subtrees
-l, --long include object size
The blob size in this example is ‘16067’. The disadvantage of this solution is that git ls-tree can process only one revision at once.
You can use instead git cat-file --batch-check < <list-of-objects> instead, feeding it blob identifiers. If location of file didn’t change through history (file was not moved), you can use git rev-list <starting-point> -- <path> to get list of revisions touching given path, translate them into names of blobs using <revision>:<path> extended SHA-1 syntax (see git-rev-parse manpage), and feed it to git cat-file. Example:
$ git rev-list -5 v1.6.0 -- gitweb/README | sed -e 's/$/:gitweb\/README/g' | git cat-file --batch-check 825162a0b6dce8c354de67a30abfbad94d29fdde blob 16067 6908036402ffe56c8b0cdcebdfb3dfacf84fb6f1 blob 16011 356ab7b327eb0df99c0773d68375e155dbcea0be blob 14248 8f7ea367bae72ea3ce25b10b968554f9b842fffe blob 13853 8dfe335f73c223fa0da8cd21db6227283adb95ba blob 13801