You will need some massive 5V power supplies. 10,000 leds at roughly 60 mA (all colours on) equals 600 A total. If your strips are e.g. 100 leds per meter and one meter long, you can also use a 6A power supply per strip; add a little overhead and I would use a 6.5A (or 7A depending on what's available) power supply per strip (or a 13A for two strips).
Depending on the length of the strip, you will have to feed it from both sides (and maybe even in the center or every meter depending on how many leds per meter; this is a little outside my area of expertise).
You will need an Arduino with sufficient RAM to address at least one strip in one go. If your longest strip is e.g. 200 leds, that means at least 200x3 equals 600 bytes; if your longest strip is 400 leds, it will be 1200 bytes and with an 328P based board 1200 bytes might become critical.
Note that longest strip here means either one strip of e.g. 200 leds or two strips of 100 leds in series. With 200 leds per strip, you will need 10,000 / 200 equals 50 pins to control them. There are probably ways to use a multiplexer to select on of the strips before sending the data but I haven't looked at that.
If you have certain patterns (e.g. one led on in every strip, dripping down from ceiling to floor), they can be calculated on the fly which will not cost much memory. Other patterns you might want to store in non-volatile memory (with 10k leds, you will more than likely need an SD card for that).
If you opt for a 3.3V board (e.g. MKR series), you will need a means of level conversion from the 3.3V (data) output of the processor to the 5V data input of the strip. You will also need capacitors at the beginning of each strip (over the 5V power supply, 1000 uF) and resistors (200..500 ohm) at the data input of each strip that receives a signal from the processor.
As the others said, start small.
I'm working on a smaller version (7 strips @ 60 leds) and decided to use a Mega so I will not run out of memory too easily. Based on layout of my strips, I will use 7 data pins (one for each strip); if I run out of storage space for patterns, I can add external storage.
I'm using two 18A power supplies, one for 4 strips and one for 3 strips.
I'm still assembling the hardware at this stage.