I wonder if it's possible to make a fully-walkable city modelled after a college campus. Let's play with some numbers. I'll make some conservative estimates of space requirements.

Indoor residential space: 400 square feet per occupant + 200 sf fudge factor
Indoor retail space: 25 sf per-capita + 25 sf margin
Indoor office/misc job space: 200 sf per office worker + 50 sf margin
Indoor schooling/daycare: 150 sf per student + 50 sf margin
Indoor supporting infrastructure: 20 sf per capita
> Hospital requirement is only around 1 sf, so I'm just adding an arbitrary margin to account for rarely-visited indoor locations (e.g. churches, community centers), storage for city vehicles (fire department), waste management, etc.

Rough population proportions by age group in U.S.:

0-18: 24% ("student age")
19-64: 63% ("working age")
>65: 13% ("retirement age")

Let's take all of the worst-case numbers, that every working-age person requires office space. The total space requirement for a population of 1000 individuals is:

Residential + Retail + Office + School + Infra
[600 * 1000] + [50 * 1000] + [250 * (1000 * 0.63)] + [200 * (1000 * 0.24)] + [20 * 1000]
600k (Residential) + 50k (Retail) + 157.5k (Office) + 48k (School) + 20k (Infra)
875.5k square feet per thousand residents

Or, 875.5 square feet of indoor space per person. This is largely dominated by residential and office space, which are highly favorable for high density construction.

Now, let's consider what a "walkable distance" is for an average person. I'm very happy to walk anywhere within a 10-minute radius (1/2 mile), but can probably live with 20-30 minutes if there is good bike infrastructure. For now let's just discuss the walking case. A 1/2 mile radius is 0.78 square miles, so a perfectly walkable space would contain all of the services I need within 0.78 square miles, or 21.7 million square feet.

How many people can we fit? Filling all of this space with single-story structures would allow us to fit 25,000 people. There are some obvious adjustments we need to make. First, we can realistically build 3 or 4-story structures for residential and office space. Next, we have to adjust for necessary road infrastructure (emergency services, delivery, waste management, etc). New York City dedicates around 50% of its land to roads and open spaces. Using suburban road densities, we can spend 10% of our land on simple 2-way streets. The remaining 90% can be spent on buildings and outdoor space. Let's say we dedicate 30% of the total land to buildings (6.5 million sf), which leaves 60% outdoor space (13 million sf). This is a fairly comfortable estimate based on looking at a map of my college campus. How many people can we fit? Let's assume some densities:

Residential: 4-story buildings
Retail: 1-story (could possibly go 2, but makes delivery logistics harder)
Office: 4-story
School: 2-story
Infra: 1-story

The footprint requirements per thousand people are:

600k/4 + 50k + 157.5k/4 + 48k/2 + 20k = 284k square feet per thousand residents

This means our 60% open, fully-walkable space can support a bit over 23,000 people, for a population density of 29k per square mile. For reference, this is slightly higher than New York City.

So, what happens when we bring parking space back into the mix? A 150-car parking lot requires an acre of space, or about 300 sf of parking lot per car. Imagine your city needs two parking spaces per car (one for home, one for work). Even if each family of 4 owns just one car (this seems like quite an underestimate), the parking requirement for our city of 23,000 people is 11,500 spaces. This amounts to 3.45 million square feet of pavement alone, just for parking! In other words, 0.123 square miles of our 0.47 sq.mi. walkable area is now parking lot. Now, how much extra road do we need to facilitate moving all of these vehicles? I'd estimate we need to dedicate at least as much extra space for driving as we do for parking, so now 0.25 square miles (53%) of our outdoor space is now dedicated to personal vehicles. Sure, Amdahl's law of parking infrastructure allows us to reclaim half of this space back if we decide to build multi-story parking structures or hide them underground, but this is a huge waste of resources considering that our city doesn't even need passenger transport of any kind!

In summary: Using some very conservative estimates and medium density, a walkable circular area of 1/2 mile radius (0.78 square miles) can quite comfortably fit 23,000 people with adequate space for residence, office, retail, schools, and remaining city infra. If we dedicate 10% of the land for minimum-necessary roads (emergency services + transportation of goods and waste), we still have 60% of the land remaining for green open space and walking. If we want to add cars, we easily lose over half of this walkable space to parking and additional road. In all of these estimates, I tried to favor cars as much as possible.