Only in Japan
In fact there are plenty of handheld computers being made and marketed today. All you need do to get hold of one is visit a consumer electronics shop... in Japan.
Here they're called "denshi jisho", or electronic dictionaries. Like the case of "computer", the name hasn't accurately represented what the devices can do for a long time now. Some still sport monochrome screens, but there are many that have color displays and enough processing power to play videos. Extras include integrated digital TV receptor, touchscreen and Wi-Fi. Most double as personal organizers, and all have some sort of mechanical keyboard, the notebook-alike clamshell being the most popular form-factor.
Could these ever be successful outside of Japan? I dunno. Smartphones have already assimilated all their functions plus making phone calls, playing games and taking pictures; in fact I'd reckon it's only the conservatism of Japanese consumers that continues to make them viable here (accordingly, most models come from traditional Japanese companies, such as Sharp and Casio). I guess that much as we may lament it, unless someone comes up with a truly fresh take on the form-factor, the age of the handheld PC is pretty much over. All hail the handheld fondleslab (aka the smartphone).