There are real-world examples for what you have in mind.
Take France, circa 1995. That's just 22 years ago now. The following is a summary from 1999, courtesy of The Register.
Until 1996 anyone wishing to encrypt any document had to first receive an official sanction or risk fines from F6000 to F500,000 (\$1000 to \$89,300) and a 2-6 month jail term. Right now, apart from a handful of exemptions, any unauthorised use of encryption software is illegal. Encryption software can be used by anyone, but only if it's very easy to break.
(In current terms, the fines ranged from €914 to €76,224, not adjusted for inflation and based on the exchange rate of 6.55957 franc per euro.)
In a society where encryption, or even just encryption that's strong enough to make a significant difference, exists but is illegal to use, it's very hard to protect digital information that is in transit. This means that information that we might readily send over the Internet encrypted might need to be delivered in person, perhaps by a trusted courier, to prevent it from being intercepted.
If your society also prohibits the use of digital signatures and similar schemes, you also have a hard time proving or confirming the authenticity of information that has been transmitted or stored electronically. This for example means that it is very difficult to enter into an agreement electronically, because whatever is being transmitted between the parties could be tampered with and nobody would be able to prove that what was finally agreed to was what was intended to be agreed to.
If your society also prohibits the use of things like hash functions then you have a big job in defining exactly where to draw the line. A hash function, in somewhat simple terms, is a mathematical function that accepts an input of arbitrary length, and outputs a value, typically of fixed length, which depends only on the input data. However, many functions with legitimate non-cryptographic uses, such as cyclic redundancy check (CRC) and forward error correction (FEC) codes can also meet such a definition. You have also pretty much outlawed every conceivable secure password verification scheme, because in order to be secure, you typically store only the output of a hash function given the password as input. (Before someone takes the preceding sentence and homebrews their own password storage: there are many, many, many nuances to this. For example, a plain cryptographic hash of a password is almost never secure enough against reasonable adversaries. Like all other cryptography, don't invent your own.)
This boils down to that it is very difficult to securely store or transmit information or agree to anything, except in person.
- Web shops and other forms of mail order exist, but electronic payment via tokens that need to be kept secret (such as credit card numbers and CVVs) is much more involved and might not exist.
- Knowing who sent an electronic message is nearly impossible, as is knowing that it has not been tampered with.
- For any kind of sensitive information, in-person delivery is likely to be preferred over any remote means of delivery.
However, I disagree with o.m.'s statement that for example cell phones would be unavailable. There are plenty of precedents for technology being available in controlled products, such as cellphones, which is not available in "raw" form. For example, even if you had the technical expertise, it's unlikely that you'd be allowed to build a cell phone yourself from scratch and use it with a random cell phone network, because a cell phone operates in frequency bands which are restricted to type-approved equipment, and getting type approval can easily require you to jump through any number of hoops. One such hoop could conceivably be installing a government-mandated back door to the encryption, or maybe the cell phone network requires that you use weak cryptography enabling the government to eavesdrop. (In GSM, A5/1 and A5/2 are the real things, and both have been successfully broken.) Broadly speaking, the only part of the radio spectrum where prior type approval is not required is that allocated to amateur radio, where encryption is prohibited.
And of course, in the words of Philip R. Zimmermann, the inventor of PGP, in volume 1 of the PGP User's Guide, the below being from the copy for version 2.6.2 released in October 1994,
If privacy is outlawed, only outlaws will have privacy. Intelligence
agencies have access to good cryptographic technology. So do the big
arms and drug traffickers. So do defense contractors, oil companies,
and other corporate giants. But ordinary people and grassroots
political organizations mostly have not had access to affordable
"military grade" public-key cryptographic technology. Until now.
People who are already violating the law tend to either not care if their communications are intercepted, or not care about breaking one more law. Laws prohibiting cryptography won't stop them from using cryptographic tools if they want to, but those laws will prevent generally law-abiding citizens from using cryptographic tools to protect the privacy and integrity of their data.