Using a cache cleaner, I freed up some cache, netting me a little over 130 MB free space in the app storage.
I then updated, and it worked. Freeing the cache again after the update, I only had 100 MB left. I also saw that the cache for the Play Store was about 28 MB, so apparently that's the size of the download for the Chrome update.
Using a disk analyzer, I found out that Chrome (after the update) uses over 100 MB for the apk (103.5 MB).
What I think happened, is that it downloaded about 28 MB, then unpacked that to 103 MB, then replaced the existing apk with that. So I needed 28 MB + 103 MB = 131 MB to do the update.
To be honest, I think that's quite a lot. I'm not sure if I'll be able to do the next one.
Also, as noted in my other question, Chrome uses quite a lot of data as well. It was at 25 MB when I asked that question, it's now a whopping 28 MB. I have no idea what's in there and no clue how to clean that other than deleting it all, which probably isn't a good idea.