According to their figures on the Falcon overview page, the Falcon V is projected to be able to launch a 4200 KG payload into a 200 KM 28 degree inclination orbit. If you're wanting to match the ISS orbit (400 KM and 51 degrees inclination), the payload is projected to be 3570 KG. They hope the actual figures will be higher.

This mass compares most closely to the Gemini spacecraft, which had a mass of 3851 KG. I know that modern materials can save you several percent of vehicle dry mass compared to Gemini. You could save quite a bit of mass using modern avionics, but the larger crew size would require more life support consumables unless you cut the mission pretty short.

Here are the Gemini numbers from the linked article:

Spacecraft Module: Gemini RM. Other Designations: Reentry Module.
Crew Size: 2.
Length: 3.35 m. Basic Diameter: 2.32 m. Maximum Diameter: 2.32 m. Habitable Volume: 2.55 m3.
Mass: 1,983 kg. Structure Mass: 638 kg. Heat Shield Mass: 144 kg.
Reaction Control System: 133 kg. Recovery Equipment: 98 kg.
Navigation Equipment: 63 kg. Telemetry Equipment: 51 kg.
Electrical Equipment: 126 kg. Communications Systems: 26 kg.
Crew Seats and Provisions: 426 kg. Crew mass: 144 kg. Miscellaneous Contingency: 100 kg.
RCS Fine No x Thrust: 16 x10kgf. RCS Propellants: N2O4/MMH. RCS Isp: 283 sec.
RCS Impulse: 9,226.19 kgf-sec. Main Engine Propellants: N2O4/MMH.
Main Engine Propellants: 33 kg. Main Engine Isp: 283 sec. L/D Hypersonic: .16.
Electrical System: Batteries. Electric System: 4.0 kWh. Battery: 180.0 Ah.

Spacecraft Module: Gemini AM. Other Designations: Adapter Module.
Length: 0.92 m. Basic Diameter: 2.59 m. Maximum Diameter: 2.59 m. Mass: 591 kg. Structure Mass: 160 kg. Reaction Control System: 200 kg. RCS Coarse No x Thrust: 6 x 43kgf. RCS Isp: 273 sec. Main Engine: 131 kg. Main Engine Propellants: Solid. Main Engine Propellants: 100 kg. Main Engine Isp: 255 sec. Spacecraft delta v: 101 m/s.

Spacecraft Module: Gemini EM. Other Designations: Equipment Module.
Length: 1.40 m. Basic Diameter: 3.05 m. Maximum Diameter: 3.05 m. Mass: 1,277 kg. Structure Mass: 250 kg. Reaction Control System: 60 kg. Telemetry Equipment: 40 kg. Electrical Equipment: 294 kg. Miscellaneous Contingency: 75 kg. Environmental Control System: 117 kg. RCS Fine No x Thrust: 8 x10kgf. RCS Propellants: N2O4/MMH. RCS Isp: 288 sec. RCS Impulse: 109,876.87 kgf-sec. Main Engine: 120 kg. Main Engine Propellants: N2O4/MMH. Main Engine Propellants: 322 kg. Main Engine Isp: 273 sec. Spacecraft delta v: 222 m/s. Electrical System: Fuel Cells. Electric System: 2.16 average kW. Electric System: 151.0 kWh.

Looking at those numbers, I do see some areas for substancial mass reductions, especially in areas like navigation equipment, telemetry equipment, electrical equipment, communications systems, and propellants. Using modern technology, you could probably save over 200 KG on the electronics alone. In the end, it seems you could save enough vehicle mass to carry 3 or perhaps 4 people on a Falcon V booster. It seems 5 would be a stretch, but not impossible.

The listed price for a Falcon V launch is $12 million. When you add the cost of the additional reduncancy to man-rate the booster, the price would be somewhat higher. Add in the cost of preparing the spacecraft (assuming it is reusable) and I'd be surprised if the mission price could be less than $20 million. Divide that by 2 or 3 paying passengers and the per seat price is still in the range of $7-10 million.

There may be other alternatives that would make the numbers work out better. For example, strap-ons could significantly increase the payload capacity and allow for more paying passengers. If the intended mission length is short enough, then the consumables mass could be kept to a minimum and the avionics would be as simple as possible. Remember, the Gemini missions carried two men for up to 2 weeks, needed significant on-orbit maneuvering capability for things like rendezvous, and used avionics and instruments that were many times heavier than what we can do today with glass cockpit technology. Between glass cockpit technology and cutting down on the maneuvering propellant requirements, you might be able to carry an additional person.

Sounds like an exciting project.