The rest of us would have to use intermediate parties that would handle everything for us.
Yes, mining it yourself is not feasible so you do need one to get ahold of bitcoin and to get fiat currency back from bitcoin, but you don't need (and maybe shouldn't use) one just for storing bitcoin, paying with bitcoin, receiving bitcoin. That's the whole damned point of bitcoin, but then people go and store it with mtgox anyway even though they're not about to exchange it, and are bummed out when mtgox loses their money. It sucks, but it's funny that half the point of bitcoin is making this situation impossible, if you bother to do it right
(Of course, if you trust a third party more than you trust your own ability to not lose/forget your private key, then I guess it makes sense, but you're trading a lot off in exchange for that)
The blockchain is a bit large at 80GB but you don't need to download the whole blockchain to run a wallet and make transactions there are "light wallets" that will do this, and currently in testing is an even more efficient method of doing this called segregated witness.
You don't need to mine bitcoin to use it. It is still very cheap to use bitcoin still as an independent node. You can use those gateways to make your life a bit easier, but from cost perspective I think they are not needed.
Now you can still get into running a node and even doing mining however as time passes the barrier to entry increases exponentially and considerably faster than the capacity of individual hardware so you can't even really catch up.
While bandwidth and storage costs exponentially decrease.
Handling the entirety of VISA's volume with bitcoin transactions would require at least a 4Mbit/s connection (about double for peak times). That's not exactly out of reach of normal consumers.
In that scenario the blockchain would grow bigger by 40Gb each day, or 15Tb per year. If you want to store that all, with current hard drive prices that would be $40 per month for new hard drives. Of course if you're not interested in looking up other people's random transactions, most of the data can be pruned as soon as the coins are spent elsewhere, giving you barely any blockchain size increase.
And all of those resources get cheaper all the time.
Resource requirements aren't a fundamental problem for bitcoin. The block size that limits it to ~0.3% of VISA's volume, and the mining system that favours a near-monopoly for the person with the lowest operating costs are far bigger problems.
Visa alone processes between 2000 and 10000 transactions a second.
Bitcoin is architected to handle 7 transactions per second. From my understanding this limitation can not be overcome by adding more miners, it is by design.