Google Nexus 5 :: Rebooted The Phone A Popup Came Up Entire Time

Yesterday I updated my phone to CyanogenMod(CM)13.0 which went fine. However when I rebooted the phone a popup came up the entire time telling me that google play services stopped working. In an attempt to fix it, I accidently removed google play services, causing a bunch of trouble at the moment.

I went to the CM Wiki and first I downloaded the "nano" variant of the GAPPS, trying to fix everything in the end. I put the file on my phone and tried to install it via recovery. This message came up however.Insufficnt storage space available in system partition. You may want to use a smaller GAPPS package or consider removing some apps using GAPPS-config ( bla bla) Error code 70. As far as I can see, it is able to mount /system and /data, but the partition does not seem to have enough space left, at least so it says.

After that I tried the same with the "pico" variant, which also gave the same result. Pretty strange for a package that small.So after taking a backup of important files I tried to wipe data/factory reset (was going to reinstall everything), to which I got an error "data wipe failed", which is correct since I can still boot in CM13 after that. So I'm a bit in trouble here. I cannot seem to wipe my phone entirely, nor am I able to reinstall GAPPS due to this error 70. Could this be a permissions problem somehow? Things I have done so far to check what the problem could be:

- I've checked the space left on the /system partition which is 501 MB. Should be enough I guess?
- I've checked the /system partition for errors via adb shell while the phone was in recovery mode, which came out clean.