I go through the math here. Admittedly I use a lot of very rough estimation. I caught some mistakes in my math in the video which I have edited here, as well as adding a section below regarding the contents of the Aron.
It's important to note that according to R' Meir, the amos used are 6 tefachim each, while according to R' Yehudah, the amos used are 5 tefachim each. (All sources used to justify these and other numbers later are cited at the bottom.)
Making these conversions, according to R' Meir:
- The outer gold box was (9.25x9.25x15.25)-(9x9x15)=89.83 cubic tefachim
- The inner gold box was (8.5x8.5x14)-(8.25-8.25x13.75)=75.64 cubic tefachim
- The cover was 1x9x15=135 cubic tefachim
- The crown I assume was one fingerbreadth thick, giving a volume of 2x(0.25x0.25x9)+2x(0.25x0.25x15)=3 cubic tefachim
- The plating of the poles was approximately 2x(3x0.5x0.25)=0.75 cubic tefachim, rounding π=3
- The rings were 4x(3x0.1252)x(2x3x0.625)=0.19 cubic tefachim
- The Keruvim, estimating based on their wingspan covering the entire Aron and scaling a human child's volume down to a height of 10 tefachim, were approximately 2x(0.56x0.56x10)+(0.25x9x15)=40.02 cubic tefachim
All of the above was made of gold, yielding a total of 344.43 cubic tefachim of gold according to R' Meir.
According to R' Yehudah:
- The outer gold box was (7.75x7.75x12.75)-(7.5x7.5x12.5)=62.67 cubic tefachim
- The inner gold box was (7.25x7.25x11.5)-(7x7x11.25)=53.22 cubic tefachim
- The cover was 1x7.5x12.5=93.75 cubic tefachim
- The crown, making the same assumptions, was about 2x(0.25x.025x7.5)+2x(0.25x0.25x12.5)=2.5 cubic tefachim
- The plating of the poles was approximately 2x3x0.5x0.25=0.75 cubic tefachim (same as R' Meir)
- The rings were 4x(3x0.1252)x(2x3x0.625)=0.19 cubic tefachim (same as R' Meir)
- The Keruvim, using the same methodology as by R' Meir, were approximately 2x(0.67x0.67x10)+(0.25x7.5x12.5)=32.42 cubic tefachim
For a total of 245.5 cubic tefachim of gold according to R' Yehudah.
The poles and middle box were both made of wood.
According to R' Meir:
- The poles were 2x3x.52x60=90 cubic tefachim
- The middle box was (9x9x15)-(8.5x8.5x14)=203.5 cubic tefachim
Totaling 293.5 cubic tefachim of wood.
According to R' Yehudah:
- The poles were 2x3x.52x60=90 cubic tefachim (same as R' Meir, as R' Yehudah agrees to a 6-tefach amah for the building, and we derive the length of 10 amos from the building)
- The middle box was (7.5x7.5x12.5)-(7.25x7x25x11.5)=98.66 cubic tefachim
Totaling 188.66 cubic tefachim of wood.
Converting to Metric and the Final Answer
According to Rav Moshe, a cubic tefach is equivalent to 751.83 cubic centimeters. Since the density of gold is 19.32 g/cm3, R' Meir's 344.43 cubic tefachim (258,952.81 cubic centimeters) of gold weighs 5,002.97 kilograms, and R' Yehudah's 245.5 cubic tefachim (184,574.265 cubic centimeters) of gold weighs 3,565.97 kilograms.
The wood used was specifically cedar wood, which has a density of 0.58 g/cm3. R' Meir's 293.5 cubic tefachim of wood weighs 127.98 kilograms, and R' Yehudah's 173.66 cubic tefachim of wood weighs 109.42 kilograms.
Adding these together yields that:
- According to R' Meir, an empty Aron weighed 5,130.1 kilograms.
- According to R' Yehudah, an empty Aron weighed 3,675 kilograms.
Contents of the Aron
As I noted, I discuss an empty Aron, whereas you include the Luchos in your calculations. The following additions are not discussed in the linked video.
The Aron contained:
- At least one set of Luchos (which were 6x6x3=108 cubic tefachim for each set) made of solid sapphire (3.98g/cm3), yielding 323.49 kilograms each. According to some opinions, the shattered Luchos were also kept inside the Aron, rather than in a separate one, which doubles this number.
- According to R' Meir, a Sefer Torah. (I'll leave its weight as an exercise to the reader; it was 2 tefachim wide by 6 tefachim tall. According to R' Yehudah, it was kept with the Aron, but not in it.)
- According to R' Yehudah, silver rods (two at 1 tefach wide by 7 tefachim long = 5.25 cubic tefachim at 10.49g/cm3 yields 41.41 kilograms)
- A jar of mann, a jar of anointing oil, and Aharon's staff were all placed next to the Aron, so their weights don't count for our purposes.
Four metric tons (=4000 kilograms) isn't a bad estimate...for an empty Aron. For a full one, you're waaaay off.
- Eruvin 14a-b - justifies π=3
- Bava Basra 14a for most of the numbers mentioned, as well as the dispute between R' Meir and R' Yehudah
- Tosfos to Menachos 98b regarding the rings
- Yoma 72b regarding the nested boxes
- Sukkah 5a-b regarding the cover and the Keruvim