Is it possible to record a range of radio frequencies in one go?
You've got two options: wideband or baseband.

Wideband

The "simple to describe but hard to build" way is to just capture the data outright and record it. The limits are bandwidth and dynamic range.

Bandwidth

For wideband, you'll need to sample at a rate of over twice your highest frequency. Considering the US FM radio band, which extends from 88 to 108 MHz, you'll need to sample at over 216 Msample/s. It's going to be very hard to find an ADC that can sample more than about 14 or 16 bits at that rate.

Also, data storage will be tough: at 16 bits/sample, you're generating 432 MB/s (3456 Mb/s or 3.37 Gb/s) of data. Note that this is faster than the peak (unrealistic) 3.0 Gb/s of SATA II and only achievable with highly tuned RAID 0 arrays or flash drives. One hour (3600 seconds) requires about 2 TB of storage.

Dynamic Range

Since each channel is generated by a different transmitter, the received signal strength will be different for each. Since you don't want a receiver-per-channel system, then your receiver has to set its gain to ensure that the closest (strongest) station does not drown out the others.

Since 16 bits is only 96 dB or so, and signal strength drops something like 6 dB per range doubling (r^-2), not counting path losses such as rain or intervening structures.  Distant stations will be received with less resolution.

Baseband

The other option is to shift the band (i.e. heterodyne) to a lower band so the total bandwidth isn't so bad.

Bandwidth

I'd recommend what is called a "Low-IF" system, where the band is shifted down close to but not quite to 0 (DC). This allows you to keep a DC blocking filter (and maybe even a 60 Hz power line filter) which will help reduce interference and circuit errors such as offset voltages.

So, for example, you shift the 20 MHz of bandwidth between 88-108 MHz down to around 3-23 MHz. This lets you sample at 50 MHz, which might let you find an 18 bit ADC, adding about 12 dB to dynamic range... still not much.

What You Lose

A typical FM radio receiver has several stages that you have to throw away in order to receive the entire band:


A tuner that selects the desired channel
A channel filter that strongly rejects interference from other channels. This filter allows high fidelity even with low-dynamic range ADCs such as 8 or 10-bit models.
An automatic gain control circuit that sets the receiver gain based on the received signal level, maximizing usage of the available dynamic range
A much smaller bandwidth (200 kHz) that you must decode
Multiple-stage heterodyne structures that let you filter the channel more than once. For example, most radios bandshift the selected channel down to a center frequency of 10.7 MHz, then filter it with a high-precision ceramic-resonator filter, then bandshift it down to 455 kHz and filter it again before creating a quadrature channel and detecting the signal.
Better control of frequencies, so that the signal from one stage doesn't create a false signal in another.


So, in conclusion, while what you ask for is not impossible, there are several things about it that will compromise performance to the point of making it nearly useless.