Multi-cam live acquisition edit setup

I need put together a 5 camera, 2 mic shooting setup for an instructional video. Cameras need not have a record function if there's a way to record multiple video and audio feeds with the editing software. I'm looking for suggestions on which editing software for mac can accomplish this. Also suggestions on what kind of additional hardware I might need. I'd like to record 24p if possible. Is Premiere a good solution? Or Final Cut? Can either of these acquire multiple live video streams? Can this be done with USB cameras? USB mics? I have access to multiple generations of macs (from the latest "trash can" to multiple generations of mac desktop towers) as well as current and legacy Premiere and Final Cut licenses. I will purchase cameras and mics for the job. I'd like to keep the camera/mic budget under $5000 USD if possible. If Premiere or Final Cut are not sufficient, I'm prepared to invest in additional software or plugins.

ALSO – I need to be able to record for one hour continuously.

I've linked a diagram of what I'd like to accomplish below. Sorry about the crappy cell phone quality.

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The destination is web video, so it doesn't have to be tv post production quality, but I'd like it to be as good a quality as I can afford.

Thanks in advance for any help!