I would say 400W with an active PFC will be just fine. Aim for Fortron, Seasonic, Chieftec..... Modern PSU will be suitable even for older setups as we talking about ATX. Care for standard molex plugs rather than the sata ones when buying new, modern ATX PSU.

I've used 300-350W PSUs on my dual P2/P3 machines in the past and none had any problem. However they were all old PSUs designed for that era when a lot of load was on +5V and +3.3V. If you want a modern PSU (which are heavily geared towards +12V loads) I don't know any particular model to suggest.

The biggest dual CPU system of that era which I ran for a few years was an HP Xeon machine which came with a 350W rated PSU from factory. It had dual Xeon 450MHz 512KB CPUs and a good amount of other hardware installed, but I think it almost always ran with a Geforce2 MX which is definitely not demanding.
Later in that machine I did try a Cascades 2MB CPU (700MHz?) and a Quadro FX2000 (FX5800 equivalent) and it did work with the same factory PSU, but I didn't run it that way very long.

An Intel L440GX+ with dual Katmai P3s which I used as a server ran for years with a 300W HiPro PSU that Newegg used to sell cheap that was really an OEM type design for Gateway or something. No video card in that though.

An IBM Intellistation with dual Katmai 550MHz, full load of RAM and 2 hard drives ran for a few years on a 350W "Tiger" branded PSU, don't know the model. Low power video card though.

Unless I'm mixing up PSUs, I think my dual socket-370 P3 1GHz machine ran with a FSP300-60PFN. Most demanding video card it ran with was a 7600GS but that was short term.
I also used an FSP300-60PFN in my Asus P2B-F desktop which was single CPU, but overclocked at 133FSB, full load of 1GB RAM at PC133CL2, and a Ti4200.
Those PSUs need recapping though. All 4 of mine contained blown cheap caps inside. Good PSUs if you can address that but if you don't want to change caps then avoid them.

I think there's some good Enermax PSUs but I don't know the model numbers. I have a 460W that turned up cheap on eBay a few years ago that I'm very fond of because it was designed in the transitional era and supports high loads on any voltage rail.

Il second the old Enermax. Iv got a dual 1ghz P3 system with 4gb of pc133 and a Fx 5950 Ultra. Used to use a recapped 300watt sparkle psu with it and a Geforce 2 mx, but a couple parts inside melted after a couple years. It still worked great!
The enermax I picked up is the Noistaker 475watt. Lots of molex, 6 pin aux connector, actually has a manual fan control as well as a sensor you can plug into a motherboard.

Even the fastest Coppermine CPUs stop below 35W TDP so if you have 2 of them you are still below a mid-tier Pentium 4. A Radeon 9800 Pro is I think around 45W (just to calculate with a high-end GPU) so a 200-250W PSU should still be more than powerful enough. So basically anything from any reliable brand will work.

@RandomStranger, not quite. I have a dual P3 system + a GeForce 3 (flashed OC @ Ti 500) and I can assure you that things are not as clear-cut as they seem. For example, this system does not start AT ALL with a Corsair VS450 (which has 16 Amps on the 3.3V and 5V rails, with a max combined wattage of 100W). If I use a video card with a 12V power connector, the system does start, however the 5V rail drops to 4.6 - 4.7V, which is a clear red flag - so I would avoid using this PSU like this for an extended period of time.
The reality is that many modern PSUs (which are designed for high 12V current) will struggle or simply refuse to start with old dual CPU PCs (or Athlon / Athlon XP PCs that rely on the 5V/3.3V rails).

@bloodem
I see. I forgot that older components can be harder an the 5V rail. I usually use more or less period correct refurbished PSUs.

To give some period correct examples
My Proliant 1600 which is dual slot 1 and 5 power hungry SCSI drives comes with a 325W PSU
My P2B-DS motherboard came out of a case with a 300W PSU

The P2B had an older Corsair 750W PSU and was happy. It now has a AX1200 and complains abut the +5v running at 4.4 System is running fine but not ideal and will reshuffle my PSU's again when I get some time

Re CPU, double check your motherboard revision. Majority of motherboards below 1.10 had a VRM that can only go as far a 600Mhz Katmai
You may be lucky and have the updated VRM, plenty of posts online about what to check and even replace if you really want. Or you can just use a slocket adapter but would have to by SMP compatable.

I definitely recommend dual PC rig, it's not really needed on a games machine but it IS loads of fun, this is mine (that Proliant also makes an appearance)

Asus P2B-DS Build

I use a Thermaltake SPG-650D-G PSU in my Dual Intel Pentium IIIE (Coppermine) system and the main consideration is that +5 needs to be able to supply at least 20 amps.

Looks like someone is selling new-old stock of these Sparkle power supplies:
https://www.newegg.com/sparkle-fsp300-60atv-3 … =9SIAD247X25294

FSP300-60ATV 300W
+3.3V@28A, +5V@30A, -5V@0.3A, +12V@15A, -12V@0.8A, +5VSB@2A

Assuming it doesn't melt like @ODWilly mnentioned sounds like something in this range would be ideal.

@chinny22 The board seller did say the BIOS was updated to take newer CPUs, and the manual I found for the board online seems to indicate it should accept later P3s in line with this. Worse comes to worse I have a 600 MHZ 100 MHz FSB PIII laying around to use.