Record every keystroke and store in a file

xinput test can report all keyboard events to the X server. On a GNU system:

xinput list |
  grep -Po 'id=\K\d+(?=.*slave\s*keyboard)' |
  xargs -P0 -n1 xinput test

If you want to get key names from the key codes, you could post-process that output with:

awk 'BEGIN{while (("xmodmap -pke" | getline) > 0) k[$2]=$4}
     {print $0 "[" k[$NF] "]"}'

Add > file.log to store in a log file. Or | tee file.log to both log and see it.

xinput queries the XinputExtension of the X server. That's as close as you're going to get as a standard (I am not aware of any standard that covers X utilities) or common command to do that. That also does not require root privileges.

If the X server and xinput support version 2 of the XinputExtension, you can use test-xi2 instead of test which gives more information, in particular the state of the modifiers (shift, ctrl, alt...). Example:

$ xinput test-xi2 --root
EVENT type 2 (KeyPress)
    device: 11 (11)
    detail: 54
    flags:
    root: 846.80/451.83
    event: 846.80/451.83
    buttons:
    modifiers: locked 0 latched 0 base 0x4 effective: 0x4
    group: locked 0 latched 0 base 0 effective: 0
    valuators:
    windows: root 0x26c event 0x26c child 0x10006e6

You can translate the keycode (in detail) to a keysym with the help of xmodmap -pke again, and the effective modifier bitmask to something more helpful with the help of xmodmap -pm. For instance:

xinput test-xi2 --root | perl -lne '
  BEGIN{$"=",";
    open X, "-|", "xmodmap -pke";
    while (<X>) {$k{$1}=$2 if /^keycode\s+(\d+) = (\w+)/}
    open X, "-|", "xmodmap -pm"; <X>;<X>;
    while (<X>) {if (/^(\w+)\s+(\w*)/){($k=$2)=~s/_[LR]$//;$m[$i++]=$k||$1}}
    close X;
  }
  if (/^EVENT type.*\((.*)\)/) {$e = $1}
  elsif (/detail: (\d+)/) {$d=$1}
  elsif (/modifiers:.*effective: (.*)/) {
    $m=$1;
    if ($e =~ /^Key/){
      my @mods;
      for (0..$#m) {push @mods, $m[$_] if (hex($m) & (1<<$_))}
      print "$e $d [$k{$d}] $m [@mods]"
    }
  }'

would output:

KeyPress 24 [q] 0x19 [Shift,Alt,Num_Lock]

when I press Shift+Alt+q when num-lock is on.

Note that you don't need to have super-user privileges to install a program. If you have write access to somewhere on the file system where execute permission is granted (your home directory, /tmp, /var/tmp...) then you can copy an xinput command from a compatible system there and execute it.

Have you considered using the script command?