Thank you very much for this synthesis of all previous answers posted before yours... Glad you are here.

Good answer, but I find the headline a little misleading - the underground civilization can determine the shape of the planet and how far from the center of the planet they are, but can't actually determine the size of their planet (only a lower bound). To say the task is trivial sort of sweeps the impossible part under the rug.

There is a major problem with big excavations when you are confined entirely underground. When you dig out a new volume, where do you put the rock and dirt you have excavated? in order for this to work, they would either need some way of disposing of the waste rock, or they would have a limited amount of free space to work with. This may be within the scope of the question, but it's unclear.

You can deduce the necessary existence of a surface simply from pressure and rock strength measurements. Tunnel walls at their given depth must be so strong to avoid collapse, the rock has not been compressed into certain exotic phases... therefore, only a finite amount of mass can exist above them.

Does this necessarily work if there’s such irregular density that a plumb line would lead them to local concentrations of mass and not a center of mass near the center of a sphere? There are presumably large air pockets and liquid water.

Or if they are at the center of mass, digging in any direction merely tells them that gravity pulls them back “down” toward their home in the center of the universe. How do they determine that they don’t live in a universe of infinite rock?

@Davislor About plumb line and mass concentrations. Geology does not work like that. You could sit on the side of the greatest mascon on Earth, and experience a deviation in gravity direction of less than 1 part in 15000. This is much less that the observational accuracy of a plumb line of spirit level. As for them starting being AT the center of the earth, that would make their observations difficult, as they are not allowed to change their elevation by much (per OP), thus would be stuck in an incredibly tiny spherical location!

@PcMan Our geology doesn’t work like that, but that wouldn’t hold true if, to take an extreme example, one hemisphere is air and the other is tungsten, would it? (Although the OP rules out anything that extreme.)

@Davislor dude, get real. please. If you choose to throw all laws of physics out the window, then nothing can be predicted or discussed. Your approach to this is just plain silly. The OP's question does include the tag "geophysics", which limits it to coherent physical laws as we know them.

variation in air refractive index due to temperature, pressure, humidity, etc. can induce curvature in light beams on the same order of magnitude as the curvature of the earth. If these aren't taken into account, they might conclude the curvature is the opposite that it is.

RE: Depth, the location of surface points should be determinable over time by triangulating lightning strikes, which should solve the problem of having to rely on surface echoing.

@RBarryYoung how does one triangulate, when you know very little about the medium through which the signal passes? There could be zones of different conductivity above them. Or an ocean. or an airless surface with no activity whatsoever. or the next layer up's inhabitants firing off their festival fireworks.

@Davislor Planets are near-spherical because at planetary scales of space (and time) even "solid" substances rearrange themselves under their own gravity to form spheres. That's why treating Earth's surface gravity as a constant and ignoring mass concentrations works to a fairly good degree of accuracy. Any hypothetical arrangement of mass to defeat that won't stay in the required shape; e.g. your solid hemisphere and gaseous hemisphere will reorganise itself into a smaller solid sphere with an atmosphere.

@PcMan It is definitely possible with enough synchronized receivers. It is essentially the same problem that geologists had when the used semisography to eventually determine the structure(s) of the earth's mantle and cores.

@RBarryYoung explain to me please how you synchronize a large number of MECHANICAL NON-ELECTRICAL seismographs? Quoth the OP:"No electricity or computers"

Note you can skip step 3 and replace it with a large enough natural tunnel network if you have enough heavy maths tomes around. One can measure gravity at any point by letting a rock fall a given distance and measuring the time. Differences in local gravity correspond to different depths=distance to the center of the planet. If the gravity differences are minuscule so are the depths differences, so they can be ignored.

@pcman Tomography. Like a CT machine. It's not complicated; you can do it with paper and pencil if you're motivated enough.

the surface may be bad at reflecting sound but earthquakes provide a HUGE impulse so bad is good enough.

Another idea how they could measure depth: They could be able to detect meteorite impacts. They might not know exactly what those loud bangs from above are, but they might be able to triangulate their positions and notice that they all originate from roughly the same distance from the core.

@Philipp again, triangulation of signals is very difficult without good time synchronization, and without electricity such synchronization is (almost) impossible to achieve. With the velocity of sound through various geology varying from 500m/s to 13000m/s depending on the geology, getting coherent measurements will be a nightmare. look at opentextbc.ca/geology/chapter/…