The random length open dipole
It is a commonly held belief among radio amateurs that an antenna has to be resonant in order to be effective at all. In practice, though, antenna resonance plays only a minor role in the efficiency of an antenna. In fact there are antenna's that are deliberately designed not to resonate on a single frequency, such as the various members of the family of traveling wave antenna's. Whip antenna's for mobile use are also non-resonant more often than not. Commercial and military antenna's are frequenty non-resonant. All these antenna's present a proper load to the transmitter only by virtue of the matching networks built into them, but the radiators themselves are essentially non-resonant. The popular 5/8 wavelength mobile whip antenna is a good example.
To re-iterate (perhaps to the howls of outrage from many radio hams who believe otherwise): there is no need whatsoever for an antenna radiator to have a length that bears a distinct, fixed relationship to the operational wavelength. It is more convenient if it does, because then we won't have to do any impedance matching, and we won't be severely punished for using a lossy coaxial feed line, but it is by no means necessary.
The misconception that an antenna has to be resonant, and that therefore its length must be a function of the operational wavelength, is largely caused by he fact that "no antenna is an island". What matters is not just what goes on in the radiator. More important is what goes on in the antenna system as a whole -- the antenna, various sections of feed line, and any impedance transforming components such as stubs, matching sections and "antenna tuners".
Let's start with that much-hallowed and much-maligned "antenna tuner" - one of the many frequently misunderstood and hotly debated components in the system. Its name is in fact a double misnomer. First, it does not tune the antenna - in order to do that, it would have to change the resonant frequency of the radiator, which is a function of mechanical length, induction, capacity and other fixed physical characteristics. Instead all a "tuner" does is to match the impedances of whatever is connected to it. It is a variable impedance transformer, no more, no less. Therefore it is more properly known as a "transmatch" which is a much better name. Second, it does not match the antenna to whatever is connected to it. It matches the antenna system (feed lines, stubs, radiators and what not) to the radio. Simply put, the transmatch is to a radio what the gearbox is to a car engine.
So. Having established that we need to look at the entire antenna system rather than just at the antenna itself, the first components we should consider are the ones that make up the feeding arrangement: the feed line and the transmatch (if any). It is here where the resonance or non-resonance of the antenna has the biggest impact. If the feed line is made from coax, then a non-resonant antenna system will not perform well: the relatively high losses in the coax will ruin the antenna system's efficiency, and the shield currents will pour large amounts of RF into the shack where it causes burnt fingers and TVI. To simplify: if there is a mismatch between the radiator's impedance and the feed line impedance, then RF energy will be reflected back and forth across the feed line until it has either been radiated by the antenna, or dissipated by the various loss factors in the system. The loss of the feed line and losses occurring in the transmatch are the most important loss factors here. If the feed line is coax (which has a relatively high loss factor) then the feed line losses will be the dominant ones, even if "low loss" coax is used!
If, on the other hand, the antenna is fed via ladder line, none of this happens. As long as it's balanced, ladder line radiates a negligible amount of RF even when severely mismatched, and its extremely low loss means that the efficiency of the antenna system as a whole is barely affected. This allows us to use an antenna that is as large as is practical (given space restrictions and the like) but the length of which is not necessarily a multiple of a quarter or half wavelength. While the old rule of "the bigger the better" still holds, that largest practical length can be a totally random length with respect to the wave length!
Of course that does not mean that the physical dimensions of an antenna are now totally irrelevant when the antenna is fed through a ladder line. The impedance of the feed line presented to the RF source (radio or transmatch) at a certain frequency is dependent on both the length of the radiator and the length of the ladder line, which in this configuration acts as a matching section. This impedance may be very high (a complex impedance of, say, 2000 + 2000j ohms is very well possible) which means that when power is applied, the voltages on the feed line can be very high. This may lead to arcing and (depending on design) excessive losses in the transmatch. Also, the efficiency of the antenna as a radiator when transmitting, and the amount of EM field it will pick up when receiving, is still a function of its length. An antenna can be non-resonant and work well, but within reason - don't expect to be able to use the proverbial knitting needle on 80 meters and get useable results! Ladder line works well, but is not a miracle "cure-all". The laws of physics still hold, and bigger is still better.
That said, a smaller antenna that fits within what little space you may have, is a lot better than a larger one that doesn't! If all you have room for is an antenna with an oddball length of, for example, 13.8 or 16.2 meters, or any other random number, just use what you have! As long as you don't mind tuning up your transmatch whenever you change frequencies, and the transmatch can handle the weird and wonderful impedances presented to it by the feed line, it will get you on the air very well. If the transmatch struggles to match the antenna system to your radio on some frequencies, try shortening or lengthening the antenna and/or feed line in small steps. Soon you'll have a length that will be easy to "tune up" on most (or even all) bands.
A random length dipole can also be installed in just about any configuration that size limitations make necesaary: as a V, as an inverted V, as an inverted L, or any other letter of the alphabet you care to use. Just make sure that the end points of the dipole stay well clear of the ground and of metal objects (masts, roof gutters, etc.) to prevent "end effects". And keep in mind that the voltages at the feed line terminals may be high on some frequencies!
Avoid using a balun! Connect the ladder line directly to the transmatch. The wild impedance swings that you can expect in a random length dipole and ditto feed line can make a balun very nervous: the mismatch will often cause excessive losses within the balun, especially if it is a type with a ferrite core, and saturation can easily occur. This will cause the balun to overheat from the RF energy that is dissipated in it, and the efficiency of the antenna system as a whole goes of course right out the window if that occurs. Note that most transmatches are of an unbalanced design, and if they have terminals for balanced feed lines that often means there is a balun built into the transmatch. If you decide to go the route of a random length, ladderline fed dipole, you might want to consider a balanced transmatch, or a Z-match with an extra coil winding to drive a set of balanced terminals. Many designs are available, such as the one by WB3GCK. Some experimentation may be necessary.
Recommended reading: The Lure of Ladder Line, published in QST, December 1993.
(Note that what is called "ladder line" in this article is actually window line.)