I actually wonder about this. Is it a matter of most cars being mostly empty, or is it inherently impossible to match the mass transit capacity. If we look at it as passengers carried over space required and time spent, here's how I see it:
Rail mass transit does not as efficiently use the land it's on a occupancy basis (there's not always a train on a specific square foot of track). In peak times, cars are more efficient on a vehicle basis. According to BART system facts[1], there are 107 miles of track. There are 669 cars, seating for 72 in 448 of them with each being 70 feet (with 59 of them having an additional 5 feet for a cab), and seating for 64 in 230 cars[2] (with an indeterminate car length, so I'll use the smaller listed), for a total capacity of 46,976 seated people. BART states that all cars can hold over 200 people in a "crush" load, so we'll assume 200 as the theoretical maximum, and say BART can carry 133,000 people when at peak (crush) capacity, and over the 107 miles of track, that gives us a density of 1,250 people per linear track mile, but with only 8.3% track utilization at any one time.
Cars do not as efficiently pack people per vehicle usually, but can more efficiently use the roads on a per-vehicle basis. Assuming very heavy traffic which is not stop-and-go, so perhaps 35 miles an hour average (the same as BART), and a 4-lane highway (two each direction), if each car is allowing two car lengths between itself and the car in front (slow traffic), we have approximately 33% road utilization (or 25%, or 20% depending on what you think the average space between vehicles is). Since carpooling seems to be at about 10% currently carpooling[4] (ignoring that it may be different in certain arterial routes, as we are discussing), we have around 1.066 people per car[5] as a lower bound. With an average car length of 177.2 inches[6], or 14.77 feet, we can estimate the people per mile on the highway during this time as being 604 people per mile of 4-lane highway.
Interesting take-aways for me:
While 4-lane highways may take more room than rail (not sure the actual sizes here), they are also more versatile.
If the highway bogs down below 35 mph, it's then less than the average rate of BART, and we need to start computing people over time instead of just people over distance.
BART has much more room to increase track utilization, but there is likely unaccounted for overhead here on each train. Optimal usage at current speed is one train arriving immediately after the prior one leaving, at 35 mph exact speed and 20 second stops, for a train of six cars (?) and 425 feet, that would be cars 2.42 train lengths apart, and a utilization of 29%, or roughly a 4x increase over current rates, 5,000 people per track-mile.
Cars have much more room to increase vehicle utilization. If we replaced 50% of vehicles with full size vans transporting on average 7 people each and didn't touch road utilization, we would be at an overall average of 4 people per car, and 2600 people per highway mile. Interestingly, if we somehow moved towards a system where smaller vehicles picked up and shuttled people with small amounts of sharing to bus-stations where they were sorted into smaller buses (40 people) going specific area depots, and from those depots dispersed again to final destinations using individual cars with small amounts of sharing, we might easily surpass rail transit systems. averaging 20.5 people per vehicle, but with somewhat more area used should put us close to 10,000 people per highway mile.
Of course, there's a lot of assumptions in all the numbers, and some speculation in the possibilities, but I thought it was interesting to figure out. ;)
1: http://www.bart.gov/about/history/facts
2: I know the total car numbers don't add up. Complain to BART, it's their data.
3: (66970 feet + 695 feet)/(7 * miles * 5280 feet/mile) = (47125 feet)/(564960 feet) = 8.3%
4: https://www.census.gov/prod/2011pubs/acs-15.pdf, table 1.
5: 105,476 drove alone, 13,917 carpooled, if we assume all carpooling was just two people per car, we get person to car density by (105,476+13,917/2)/105,476 = 1.066 people per car
6: https://www.reference.com/vehicles/average-length-car-2e8538...
7: 5280 feet/mile / 14.77 feet/car * 0.33 highway mile utilization = 117 cars at highway lane mile utilization. 117 cars * 1.066 people/car = 126 people per mile of highway lane. 4 lanes fives us 604 people per highway.