The simple answer to that is: no, but it's fun.
The slightly less-simple answer to that is no, but the number of science fiction authors, even 'hard' SF authors, who don't try to explore what an FTL-less universe would be like is surprising.
According to Einstein and backed up by many theories since, the speed of light is the absolute maximum velocity that any object within our universe that possesses mass can travel at. This speed is just short of a startling 300,000km per second, but interstellar and even interplanetary distances dwarf this number. Travelling to the moon at the speed of light would still take 1.3 seconds, for example, whilst the Sun is 8 minutes, 19 seconds away. Travelling to our nearest interstellar neighbour, Alpha Centauri, would take about four and a half years. It would take 100,000 years to cross our galaxy from one side to the other and two million to travel to Andromeda. Compared to the size of the universe, that's still mucking around in our back yard.
For this reason, most works of science fiction employ a faster-than-light (FTL) drive which circumvents the lightspeed restriction. The name is actually a misnomer, as simply accelerating past the speed of light is impossible (it would require infinite energy and would also result in time going into reverse for the traveller, creating significant paradoxes). Most FTL 'cheats' by allowing the traveller to suspend the rules of relativity by instantly teleporting from one point to another by way of wormholes (used in Dune or Peter F. Hamilton's work), or by 'warping' space so the laws of physics no longer apply (this is the favoured approach in Star Trek). Another approach is to have spacecraft move into a parallel universe which is either much smaller than our own but where every point corresponds to a point in our universe. This approach is favoured in Babylon 5, where a ship enters hyperspace, travels a few hundred thousand kilometres, and returns to our universe several light-years from where it started out. Warhammer 40,000 uses a similar realm known as the Immaterium (popularly called the Warp) although the difference is scale is not so extreme, where journeys between stars a few light-years apart can still take days or weeks, whilst traversing the entire Galaxy takes several years. Star Wars mixes the two approaches by having a ship accelerate in real space to lightspeed and is then blasted into hyperspace by the acceleration.
All of these approaches and many others are interesting, but tend to ignore a very interesting feature of real-life physics, namely time dilation.
As a spacecraft approaches the speed of light, relative time on board the spacecraft slows down compared to the outside universe. For example, a spacecraft that travels to Alpha Centauri and back again at 90% of the speed of light would appear, from the POV of people on Earth, to take roughly nine years to complete its trip. From the POV of people aboard the spacecraft, however, it would take just a few weeks. As lightspeed is approached, time aboard the spacecraft slows to the point where immense journeys that take thousands or even millions of years from the perspective of the outside universe are achievable in just a few years of on-board travel.
In fact, time dilation can create incredibly warped effects. A ship that left Earth and could constantly accelerate at 1G would achieve 99.99999% of lightspeed and, as a result, could reach the edge of the observable universe (about 13.5 billion light-years away) in less than a century of on-board time, i.e. within a human lifespan. Of course, this would require a fantastically advanced space drive and some mechanism to prevent the ship from exploding the second it hit any interstellar debris larger than a pinhead, but it is certainly feasible within the laws of physics as we currently understand them.
On a more modest scale, ships accelerating to appreciable fractions of lightspeed and flitting back and forth between nearby stars is reasonably realistic and, in fact, is the basis of interstellar travel in Alastair Reynolds's Revelation Space universe. For the stories to unfold in the Revelation Space novels, Reynolds has two storylines proceeding in tandem on different planets, but they are actually taking place in different years. For example, Storyline A is happening in AD 2500 but Storyline B is taking place in AD 2530 on a planet thirty light-years distant. At some point the characters from Storyline A get on a spacecraft equipped with a Conjoiner drive (which allows rapid acceleration to 95%-99% lightspeed) and travels to the second planet, where they arrive in AD 2530 and join in the storyline there. Thanks to time dilation, the journey only seems to take a few months, maybe a couple of years, from the perspective of the first set of characters, and they age accordingly.
Bizarrely, very few SF authors seem to pursue this way of handling FTL travel. Mainly this is because space operas love to have multiple sets of characters on multiple planets and it's important that the journeys between the planets take place very quickly and everyone remains in the same temporal space as everyone else otherwise things will get very confusing very quickly. This is a decent enough reason, and is necessary to make the stories make sense. However, it is intriguing that more stories are not written which take into account the lightspeed restriction from the outset.
Outside of Reynolds though (and Reynolds does include a back-door FTL method in the Revelation Space books, albeit a method that appears to be inaccessible for humans, and does use FTL in other books), the only author who seemed to rigorously enforce the lightspeed limitation was Arthur C. Clarke (with the sole exception of 2001: A Space Odyssey, and even there the FTL method was retconned out of existence in the sequels). His Rama Cycle of novels had an alien spacecraft travelling at a bit above half the speed of light (taking twelve years to travel the eight light-years from Earth to Sirius, for example), whilst The Songs of Distant Earth has a human-built spacecraft travelling at lower speeds with the crew in suspended animation.
Whilst the development of new physics which permits FTL travel is possible, at the moment such a drive appears flat-out impossible. While this should not restrain authors' imaginations from using FTL methodology, it is a shame that more authors do not employ time dilation as an acceptable way of getting characters between stars without dropping dead from old age as certainly that appears to be the only way of realistically carrying out interstellar travel at this time.
Pushing Ice and the Revelation Space Trilogy (Revelation Space, Redemption Ark and Absolution Gap) by Alastair Reynolds.
Rama II and The Garden of Rama by Arthur C. Clarke and Gentry Lee.
The Songs of Distant Earth by Arthur C. Clarke.