To properly answer this we need to understand a bit about the history of how telephone calls were completed in the early days. While in theory you could just run a line between two stations (a private line), this isn't very useful because a) you might want to talk to more than one station and b) it quickly becomes overwhelming if you want to implement a solution for a). To alleviate this, telephone exchanges were created. All calls that were placed were manually completed by an operator at a switchboard in the exchange.

To do this, the caller would take the receiver off the hook and manually activate a magneto that would send a ringing current down the line. This would alert the operator on your trunk appearance and they would pick up and ask you who you wanted to call. You would give that information and they would attempt to connect to the distant station through a similar mechanism on their switchboard. It's not well documented in anything I can think of as to how long they would alert the far station of the call, but it would be at most a minute or so. If the far station was answered the calls would be bridged together. If not, the operator would tell the calling party that no one was there and to try later.

Now, obviously there are scaling problems here. So relatively quickly, machines that could provide automatic switching of calls were provided. This is an incredibly detailed field and there's no way I can get into all of it here, but the first automatic switches are generally attributed to Almon Strowger. He received his first patent for them in 1891. At this time the Bell System control over the US telephone network was strong and as he didn't work for them, they didn't generally install his equipment. However, many smaller independent companies did. These systems allowed for the subscriber to call directly to another subscriber without operator intervention. They were initially not particularly complicated systems but they grew in complexity quickly.

These systems are all built in a hierarchical fashion. Your call steps through various levels of the system based on the digits that you dial. The dialing of digits causes manual switches to move to complete the electronic connection.

To get to your question, early switches did not have any sort of way of monitoring whether a call was in the ringing state for an extended period of time. However, the offices where these switches existed were generally staffed 24 hours and technicians would note whether calls were in a stuck state. Over time circuits were created to alert on long calls and to also limit the length of unanswered calls. More advanced switching systems did this by default and could be programmed to limit the length to a value that the operating company felt was appropriate. It was often assumed that ringing times of over 20 seconds would more often than not result in an unanswered call. Fully electronic systems that began appearing in the 1960s had what was called "ringing timeout" parameters that could define this.

So, in conclusion, while in some circumstances you could do this, technology was quickly developed to limit the ability.

Sources:

A History of Engineering and Science in the Bell System, Vol. 1, The Early Years, 1975, Chapter 6

Applicability of Automatic Switching to All Classes of Telephone Service, Arthur Bessey Smith, Trans. AIEE, Vol. 38, p. 1567, Dec. 1919

The Rape of Ma Bell, 1988, Kraus & Duerig, Chapter 17

Well, you have to remember that until the advent of Bell Electronic Switch Systems, in the 1960s, most of the calls were handled by a live human operator that would connect the call from the caller to the destination number through a plug switchboard. So, there was no "ringing" or "busy tone", the operator would just say something like "the destination number is not answering, sir", after a few attempts.

Before the popularization of ESS switches in the 1970s, the most advanced telephone markets in the world would use of rotary switches (electro-mechanical) to connect calls without an operator. Those entered the market in the early 1920s, 50 years after the telephone was invented. Those would disconnect the call automatically after a preset number of rings without call pick up, in order to avoid central line and telephone exchange congestion.

So, for almost 100 years after the telephone was invented, calls would not ring indefinitely as they would be, most of the time handled by a live human operator that would just tell the caller there wasn't anyone to answer the call in the destination.

Here are some good links about the history of telephony switching:

History of the Switchboard

History of Telephony Switches