Not necessarily. Oak barrels are a staple of traditional winemaking, and most red wines (and many whites) feature oak barrels in their process to some degree. However, hand-coopered white oak barrels are quite expensive, and they must be kept meticulously clean or they'll spoil. Many home winemakers avoid these hassles by using oak chips. These are produced from the same wood as barrels (from forests in America or France), but they are very cheap and easy to use. They simply need to be added to the fermentor until their flavor is sufficiently absorbed.

Eventually, most advanced winemakers invest in barrels, either alone or as part of a winemaking group. Why? Well, the effects of oak aging are more complicated than just the wood flavor imparted. Since oak barrels can "breath," a slow, complex process of evaporation and oxidation occurs during barrel aging. This lends much more complexity and roundness to the wine than can be achieved by aging in glass or stainless steel. Commercial wineries who age their wines for 2 to 3 years in old oak barrels are relying on this process, rather than on the flavoring potential of the wood.

Expanding a bit on what has already been said:

Oak barrels perform two primary functions. First they provide chemical compounds such as gallic acid, ellagic acid and vanillins which combine with compunds in the wine and add flavor and complexity. Secondly, they concentrate the wine via evaporation of water and alcohol at the rate of 3 or 4 percent per year. Regarding breathing; a strong vacuum will be created in a properly bunged barrel which indicates that a barrel exhales constantly (alcohol and water vapors), but only inhales (air) when the bung is removed.

Oak chips provide some of the first function, but because they are thin and roasted all they way through in convection ovens, they have less complexity than barrels, which have a thin toasted or charred inner layer uner an outer layer of air dried oak. Oak beans (cubes) and staves are toasted over an oak fire and because of their thickness are more similar to barrels than are chips. They tend to add greater complexity and at a much slower rate, thereby giving you more control. Of course they also cost a bit more, but in my experience they are well worth it.

It is also important to note that micro-oxygenation occurs in barrels and is instrumental in the formation of "softer" tannins. The oxygen is mostly a result of opening the barrel for sampling and racking, and of course that occurs when you rack carboys as well, so if you store on oak beans in your carboys for 12 months and rack four or five times, your "oakiness" will be similar. However, you will not be able to replicate the concentration by evaporation that occurs in the barrel.

Finally be aware that if you ferment white wines "sur lies" (on the lees), you will lose much or all of the vanilla flavor of the oak that is extracted while the lees are present.