Tape echo effects were invented towards the end of the fifties. Presumably the very first tape echo machine was invented by Charlie Watkins of London, England, in 1958. It was based on a small loop of 1/4" tape onto which audio was recorded by a magnetic head and then read by three separated heads. What was read from the tape was amplified by the internal all-valve circuit and eventually recorded back to the tape, creating the feedback or "sustaining" echo effect. The delay time of the unit was determined by distance of the write head from the read heads, in conjunction with the speed of the tape. Many manufacturers of this era produced their versions of echo machines, mostly based on the magnetic tape system. The famous Binson Echorec was based on a circular drum head with a metallic magnetic stripe. It was used by Pink Floyd at the time of their album "Echoes". Another famous tape echo machine was the Maestro Echoplex, heavily used by pianist Herbie Hancock. Other notable units were the Selmer TruVoice Echo, the Dynacord Echocord, along one of the most famous and widely used from 1973 to present day: the Roland RE-201. This particular model has a tape loop about 4 meters long which is recorded, read and erased continuously. It consists of one erase head, one write head and three pick-up heads, which when combined with the variable speed DC capstan motor allows many different delay configurations. This machine also offers a reverberation effect based on a spring reverb tank. The combination of echo and reverb gives an "ethereal" ambience effect which eventually inspired the engineers so to name it "Space Echo".
Nowadays these machines are very sought after but quite rare to find on the market, and their evaluation increases year after year. There are many aspects that make the tape echo stand out from a modern digital delay. Here are a few of the main features:
1) the unpredictability. A capstan motor is never perfect like a digital clock signal, resulting in slight variations of the delay times and pitched sound.
2) the frequency response and dynamic range. A magnetic tape is not as clean and flat as a digital recording. Especially a loop of tape which is cyclically erased and re-recorded many times in a short time lapse.
3) noise, hum, distortion. Defects that a digital system shouldn't have, sometimes are wanted and can become "musically acceptable".

There is one way only to obtain all of this. To get the real thing.
Actually there's a much affordable and easy alternative: to use an accurate digital simulation.