When I practice the cello, sometimes it is noticeable that, when playing big chords, they tend to be out of tune - the C string sounds a bit too flat. This is the issue of tuning by 5ths for any string instrument. Is there any way to tune a string instrument with well temperament without the need of a tuner, or the tedious way of bringing 5ths closer together? I'm looking for a relatively quick and efficient method of tuning only with the cello itself? (no tuners or pianos)
I play fivestring (extra low F) and what I generally do is, I first tune top-to-bottom in mostly pure (Pythagorean) fifths, then tune the F up again as much as necessary for the major-third15ma between the outermost strings to sound acceptably consonant. I then also tune the C and sometimes G strings up a bit until fifths on the low strings can be played properly. The result is actually pretty close to 12-edo†, and while I'm not a fan of 12-edo it is obviously very sensible when playing with piano/guitar etc..
In principle the same strategy also works without the F-string, by directly tuning the major sixth8va between C- and A string. If you make this a true just Ptolemaic consonance, you end up with quarter-comma meantone tuning, but I wouldn't recommend that – in my experience it's just too narrow, and Pythagorean-ish melody lines end up really clashing with harmony notes. Instead I suppose it's better to just tune every pair of strings very slightly narrower than Pythagorean. To get consistent results without counting beats (that's how piano tuners traditionally do it), it helps to always press the upper string down just ever so slightly before the bridge. You want to tune up each pair about 5ct from Pythagorean.
†To be honest, when playing live I use a 12-edo clip-on electronic tuner because it's just so much less hassle, especially in a loud environment. It's unromantic but sensible to have such a tuner (you can get them from ~5€ nowadays) permanently clipped on the bridge.
acoustic cellist dropping in here. The reason to tune the open strings to 'true' temperament is so that open double-stops will sound right. All other double or triple stops simply need to be fingered appropriately to get the harmonies desired. As mentioned in other answers and comments, finger position must be adjusted slightly to account for: string height above fingerboard, string tension, bow pressure related to sound volume, and more. You seem to be under the impression that there's one exact position for a note, and that is not the case. There are always adjustments to account for the physical mechanics of the instrument as well as matching pitches in a double stop.
For a simple example: play an open G-string or C-string from ppp to fff and watch the shift in pitch on a good chromatic tuner.
Well temperament with twelve notes per octave can be achieved on string instruments. I am a violinist and this way of tuning works for all situations I have uncounted - except stopping perfect fifths as that is not easy on the violin/ viola, but possible and easy on the cello/ double bass.
The only way to achieve this, without resorting to a piano or tuner every single time, is to learn and internalise the sound produced when all the strings are tuned exactly to the right frequency. For example with a cello: 65.41Hz (C2), 98.00Hz (G2), 146.8 Hz (D3) and 220.0Hz (A3). So, to begin with you would have to use a tuner until you can tune your instrument just by ear using 'slightly narrower fifths'. Though unless you have perfect pitch you are going to need one in tune note as a reference, most commonly 440Hz (A4).