What are some more forceful ways than a .gitignore to keep (force) files out of a repository?

Is there some way I can make it totally impossible to push certain files onto a repository?

Yep, you can use hooks like this to prevent several files to be committed.

pre-receive hook

#!/bin/sh

# Check to see if this is the first commit in the repository or not
if git rev-parse --verify HEAD >/dev/null 2>&1
then
    # We compare our changes against the previous commit
    against=HEAD^
else
    # Initial commit: diff against an empty tree object
    against=4b825dc642cb6eb9a060e54bf8d69288fbee4904
fi

# Redirect output to screen.
exec 1>&2

# Check to see if we have updated the given file
if [ $(git diff-tree -r --name-only $against | grep <ANY FILE YOU WANT TO FIND OUT HERE> ) ];
then

    # Output colors
    red='\033[0;31m';
    green='\033[0;32m';
    yellow='\033[0;33m';
    default="\033[0;m";

    # personal touch :-)
    echo "${red}"
    echo "                                         "
    echo "                   |ZZzzz                "
    echo "                   |                     "
    echo "                   |                     "
    echo "      |ZZzzz      /^\            |ZZzzz  "
    echo "      |          |~~~|           |       "
    echo "      |        |-     -|        / \      "
    echo "     /^\       |[]+    |       |^^^|     "
    echo "  |^^^^^^^|    |    +[]|       |   |     "
    echo "  |    +[]|/\/\/\/\^/\/\/\/\/|^^^^^^^|   "
    echo "  |+[]+   |~~~~~~~~~~~~~~~~~~|    +[]|   "
    echo "  |       |  []   /^\   []   |+[]+   |   "
    echo "  |   +[]+|  []  || ||  []   |   +[]+|   "
    echo "  |[]+    |      || ||       |[]+    |   "
    echo "  |_______|------------------|_______|   "
    echo "                                         "
    echo "                                         "
    echo "      ${green}You have just committed code  "
    echo "      ${red}Your code ${yellow}is bad.!!!      "
    echo "      ${red} Do not ever commit again    "
    echo "                                         "
    echo "${default}"
fi;

# set the exit code to 0 or 1 based upon your needs
# 0 = good to push
# 1 = exit without pushing.
exit 0;

---

Note:

GitHub does not support using hooks in this way.
They have their own WebHooks

In this case you can use hooks as well but on the client side.
The same code can be placed inside pre-commit hook on the client side.