After looking at a tea strainer on Google something immediately came to my mind, however it will involve perhaps demolishing a sieve.
If you happen to have an old sieve that you are willing to modify somewhat then that's great - carry on reading.
Remove the old metal mesh from your old strainer, some wire cutters would work on a stronger metal but scissors may work or you may even be able to push the metal out.
Remove the metal mesh from your sieve and cut out a square / circle or whatever of the material the same or a similar size to the piece that you have just removed from the old strainer.
Insert newly made mesh into the old casing and seal it down with some glue - leave to dry for a day to make sure that it is fully cured and won't get into your beverage.
You should now have a makeshift tea strainer thing - probably won't be as fine a mesh than the original but should suffice for most tea leaves.
Note, if you do not want to use the old casing you can probably makeshift your own out of paper clips or metal wire and shaping them into whatever shape you want it and then making a cage like thing with the sieve mesh.
Another method I found when looking for a different solution to your problem is to use some tin foil. I'll summarize the contents of the link I have attached:
- Cut a foot square of tin foil
- Fold in half twice so it is now a quarter of the original size
- Pour tea leaves into the center
- Bring corners together above the leaves and twist them together making a sort of stick - you'll end up with what looks like a tin foil ball on the end of a tin foil stick
- When you are ready to brew your tea, poke some small holes in the tin foil ball with a cocktail stick or similar and then brew in the normal way for the normal amount of time!
Here is what it should sort of look like: