One can always define a start byte as an indication that a message is beginning (and the ASCII SOH, STX, and ETX codes were intended for such purposes). However, the standard hardware and protocols for connection to data-transmission equipment (RS232C and later) operate at a lower level, and it is generally neither possible nor desirable to alter that arrangement (especially via software).
High performance synchronous data transmission schemes, such as those used on local-area networks and wide-area transmission systems do use elaborate frame markers. The frame marker is a distinct pattern of bits that never occurs in the stream for message data. There is typically a special rewriting rule that essentially "escapes" any in-data occurrence of a similar bit pattern so that transmission equipment will not see it as a frame marker. These escaped patterns are reconstructed by the recipient so the sender and receiver never have to pay attention to this. These arrangements make specialized hardware even more important, such as in the typical Network Interface Card (these days, motherboard chip) on personal computers.
BACKGROUND ON ASYNCHRONOUS SERIAL COMMUNICATION
It is useful to think of asynchronous serial transmissions as asynchronous between character/data frames and synchronous within the span of the character frame (including the start bits and initial stop/fill).
With this scheme, there is a constant fill signal between the frames and it is usually at least one data-bit wide, although some arrangements require a 1.5-bit or two-bit stop/fill. The stop "bit" uses the same signal level and can be considered the minimum fill period before another start bit will arrive.
When a frame is arriving, it is necessary to synchronize with the predetermined number of bits it is expected to carry. The transition from the fill to an opposite level signal is accomplished by the start bit which is always opposite to the stop/fill level. The sampling of the bits can be timed to happen in the middle of subsequent bit-arrival periods.
Technically, if frames were being sent at the maximum rate, it would not be necessary to send any stop/fill, proceeding to the start bit of the next frame immediately. However, counting on at least one bit worth of fill before the start-bit transition helps to keep the sender and receiver synchronized.
If you think of the asynchronous streams as being encoded from key depressions using a keyboard, you can see the importance of allowing arbitrary fill between character frames. Once it is known what frame to send next, it can be inserted immediately, with its start bit, at the agreed bit rate, after there has been at least one bit worth of preceding stop/fill.
It is also useful to notice that, in typical low-speed asynchronous transmissions, there are only two kinds of bits/levels, so the only way the presence of data as opposed to fill can be distinguished is by a marker scheme like this where the start of the frame is uniquelly detectable and the end of frame is predetermined (unless there is a more-sophisticated variable-length frame structure generally not used in asynchronous serial communication). It is actually rather difficult for a receiver to discover the bit rate of a transmitter without some additional agreement, such as looking for a recognizable data sequence from which one can estimate the bit rate which would have it arrive correctly when it arrives in incorrect form.
Even though high-speed modems now transmit complex analog signals that aren't described in terms of two simple signal levels, the RS232C (and later-mode) digital communication between a computer UART and the data coupling on the modem is pretty much as described.
High-speed modems also have additional capabilities for synchronizing with a distant end-point, as you can tell by listening to the signal audio while a connection starts up. In addition, There are separate signal lines in the serial cable to the computer that are used for pacing between the computer and the modem so that the sending party does not transmit new data frames faster than the receiving party (either computer or modem) can accept them. But a frame, once started, is always started at the agreed synchronous speed.
Wikipedia has a good description of asynchronous serial communication, what computer serial ports use.
There is a common over-simplification that suggests the stop bit determines the length of the data. That's not the case. The stop bit looks just like a level for another data bit. The way the stop bit and the period until the next start bit, are recognized is by knowing the bit rate at which in-frame data and start/stop bits are being transmitted and knowing how many bits a frame contains. Otherwise, there is no way to distinguish a stop bit from just another bit of that polarity as part of the data frame.