A growing number of devices support HD playback, and more are set to join them as HD audio becomes more widespread. The examples given below are not endorsements of a particular company's product - it's a matter of personal taste, budget and various other factors, and it depends on where and how you're intending to listen to your HD files!
If you have the files stored on your hard drive and want to listen to them via your computer, you'll need to connect your computer to a USB DAC (Digital Audio Converter) that supports HD formats, then plug that into your speakers/headphones. Quite a variety of products exist that will do this - the Arcam irDAC, the Naim DAC-V1 and the T+A DAC 8 are just three examples. Some headphone sets also have the necessary DAC built in, meaning you can bypass this step. Note that the Sonos system does not support HD audio - it only goes up to 16/44.1 (ie CD quality).
You will also need audio playing software that is capable of playing the files. Most software can do this but there are some notable exceptions, such as Apple's iTunes (which doesn't support FLAC files). Compatible programs include VLC, Winamp, Media Monkey and Songbird (Songbird is for MACs; the other programs are for PCs).
If you are an iTunes user then you can play HD files, but you'll need to convert the FLACs into Apple's equivalent format, ALAC, first. Once you've done this (the conversion is lossless), you should end up with HD ALAC files that iTunes can play with no difficulty - they'll also be playable by iPods.
If you want to listen to your music on an external storage device, you'll need some specialised equipment - a network music player and some good speakers to go with it. Various manufacturers exist - Cambridge Audio, Marantz, Linn and Sony all offer a range of network music players and accompanying speakers. You can either plug the speakers in or, increasingly, stream the music wirelessly from the storage device to speakers elsewhere. The possibilities are far too numerous to list here!
If you want to play your music on your mobile device or personal media player, then this depends what model you have. Some modern smartphones support HD audio - the Samsung Galaxy Note 3 and the LG G2, for example - but the majority do not. There also exist specialist HD personal media players - various models are available from Sony and Astell & Kern.
Finally, it's equally important that your listening environment and the rest of your equipment are suitable - there's obviously no point in listening to HD audio files on a noisy train, for example, no matter how good your device is! To get the most from HD audio, make sure your loudspeakers/headphones are of a similar standard to your DAC hardware - otherwise you won't get the benefit of the superior recording techniques.