DIY telescope to observe nebulae How can I make a good telescope that is powerful enough show nebulae and other deep sky objects ? (consider low budget build)
Absurdly cheap and (potentially) very fun!
The absolute cheapest telescope to make would be extremely hard to make, you'd be doing something for the first time, but it would at least be fun!
Make your objective lens out of ice, and find a way to polish it smooth. Don't make it as big as these though. You can then use an eyepiece out of a pair of old binoculars or a toy telescope or use a dime-store jeweler's loupe for an eyepiece.
From Has anyone ever tried to make a simple telescope using ice?
These are solar collector projects, a telescope objective would have to be polished to a much better shape.

 
Here is a functioning camera lens made from ice for example, from this answer there:
 
 
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Source:  Mathieu Stern
Chromatic abberation is a problem, so use a filter that removes blue and green (I guess you'd call it a red filter) and look at things that are mostly hydrogen.
From Alex Kozik's Orion Nebula spectral composition we can infer that the Orion Nebula would still have the shape of the Orion Nebula in the presence of chromatic aberration if got rid of green and blue light.

From James Kaler's1 The Spectrum of the Orion Nebula:

1Professor Emeritus of Astronomy, University of Illinois


But seriously folks
If we throw out any hope of an electronically driven and computer controlled equatorial mount and just want to look at things as they slowly drift then spend most of your budget on an f/8 spherical mirror, either a second hand one from a discarded telescope or one out of a catalog like an Edmund Scientific from the 1970's. Seriously I don't know if there are catalogs where you can order premade mirrors any more, but it was fun to dream!
You may have to invest in a telescope mirror making kit. There are an abundance of resources on how to make home-made mirrors, and as long as you are f/8 or longer you don't need to worry about figure correction (making it a parabola instead of the natural spherical shape it will want to have)
Next, put your mirror in a Dobsonian mount made from plywood or scrap metal, depending on your materials and particular skills and tastes.
You'll need a diagonal mirror or prism to send the light out the side and a stick or spider to hold it near the center of the aperture. If you are only going to use one eyepiece then you can find a simple way to hold it in place, but you've got to do some reading about how to align the whole thing.
That would be an excellent follow-up question here!
You can also take a cue from Sir William Herschel and instead of using a diagonal in the center of the aperture, just peer over one edge with the eyepiece and (since your telescope will be a lot smaller) use a diagonal mirror or prism attached to the edge. You'll have more distortion, but if you increase your f/ number (the one below was f/10) and stick with low magnification, you'll see some nebula, at least as opposed to not seeing them. :-)
I used a small front-surface mirror, some alligator clips, an eyepiece from an old pair of binoculars, a cardboard tube (a la Clyde Tombaugh) and a weak concave magnifying mirror for my first 10 cm Herschellian reflector telescope and had a truly thrilling time!
From here:
 
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left: Source, right: &quot;Two Dobsonian telescopes on display at Stellafane in the early 1980s.&quot; Source