The following `awk` program would do the job. It assumes the data is stored in `data.txt` in the first column (but can easily be adapted for any other column). It also assumes there are no empty columns, and only complete chunks.
awk -v cs=4 '{if ((i=NR%cs)==0) {n_ch++; i=cs};buf[i]+=$1;} END{for (i=1;i<=cs;i++) printf "%d\
",buf[i]/n_ch}' data.txt
The chunk size is passed to `awk` via the `-v cs= _size_` statement.
It will, for each line, determine the "entry number within the chunk", `i`, via `i = "line number" modulo "chunk size"`, and sum the entries into an array `buf`. Whenever one chunk is complete, the chunk counter `n_ch` is increased.
In the end, we print the average for all entry numbers.