The Bitcoin wiki page on scalability looks at what things would be like if transaction volume was similar to that of Visa. It states:
Today the Bitcoin network is restricted to a sustained rate of 7 tps
by some artificial limits. These were put in place to stop people from
ballooning the size of the block chain before the network and
community was ready for it. Once those limits are lifted, the maximum
transaction rate will go up significantly.
The page also contains various calculations on what the implications are of the Bitcoin network matching Visa's rate of transaction processing (which the page states as being around 2,000 transactions/sec). Figures on storage required for the blockchain are included in the calculations, but it's clear that as Bitcoin grows the core backend nodes will require commercial levels of investment in hardware & systems.