hmmm interesting question, let me put on my Mcguiver hat on (this might or might not work but based on my experience in screen printing and art it theoretically should).
You'll need a wood frame (picture frame might work), some very thin porous silk (or a polyester material), rubber cement, rubber cement thinner, a wide squeegee, and acrylic paint.
Step one: Build your frame and staple your silk onto it tight like a drum.
Step two: create your artwork. You will need to draw out what you want to print. Any area that is being printed will be in reverse and the areas that you do not want to apply a resist on which would be the next step.
Step three: Create your resist, this should be a thin enough rubber cement that will not bleed, but thick enough to fill in the spaces on your silk. Paint it onto the areas that you don't want ink to transfer through (you'll likely have to throw out the brush afterwards. Let it dry and you should have the makings of a silk screen. Make sure you don't have big gobs of cement or it will create ridges and may create issues.
Step four: In theory you should be able to use acrylic paints to do this. It should be pretty thick paint and not overly runny. If it is runny it might make a mess. Water based inks for actual silkscreening is pretty thick stuff (about molasses consistency). Set up on a flat surface (put down lots of newspaper to pick up on the mess) and put down a pad of newsprint to prime your screen.
Step five: Prime your screen by pouring the paint at one end of the frame in a single continuous bead. Your squeegee should be the size of the frame, but it might still work without it, but be prepared to see streaking and potential bleeding. Run the squeegee across the screen on the news print in a single continuous direction. Take the newsprint out and repeat until you get a clean impression. If there are areas that you missed with your resist solution, clean your screen with cold water, let it dry and paint your resist in the areas that didn't work.
Step six: You're ready to put your sheet down (or T-shirt if that's what you're hoping to do) and careful not to let the screen touch the materials until you're ready to run the squeegee across.
That should be it. If you want to get fancy, you should really hinge your screen onto a solid surface like a board of wood, but this is rough basic way of doing this with most standard materials you can get in the home. It's also less toxic than traditional silk screening. The only caveat in this technique is likely you've created a permanent screen. You can try to remove the resist later with cement thinner, but I think you'll have to remove the screen and create a new one. When you're done, you should always clean up your screens immediately so you can use them again. Don't let the acrylics dry on your screen, if it does, then it will likely be done and you'll have to start again. Traditional silk screening is far more complicated than this, so much of that has been simplified in this experimental tutorial.
Good luck and let me know if this works if you try it.
I also came across another home technique. This is closer to actual silk-screening so some of the materials will be harder to get.
They use a silk screen or screen printing.
Screenprinting, or serigraphy, previously known as silkscreening is a printmaking technique that traditionally creates a sharp-edged image using a stencil and a porous fabric. A screenprint or serigraph is an image created using this technique. It is related to resist dyeing on cloth.
It began as an industrial technology, and was adopted by American graphic artists in the 1930s; the Pop Art movement of the 1960s further popularized the technique. Many of Andy Warhol's most famous works, including his Campbell's Soup Cans, were created using the technique. It is currently popular both in fine arts and in commercial printing, where it is commonly used to put images on T-shirts, hats, CDs, DVDs, ceramics, glass, polyethylene, polypropylene, paper, metals, and wood.
In electronics, the term screenprinting or screenprinting legend often refers to the writing on a printed circuit board. Screenprinting may also be used in the process of etching the copper wiring on the board or computer chips.
Graphic screenprinting is widely used today to create many mass or large batch produced graphics, such as posters or display stands. Full color prints can be created by printing in CMYK (cyan, magenta, yellow and black). Screenprinting is often preferred over other processes such as dye sublimation or inkjet printing because of its low cost and ability to print on many media.