Daejeon is a very safe city by global standards. Note that Daejeon does not play host to a US military base, and as such, anti-American sentiment and by extension, anti-Westerner sentiment is low. If anything in town could be deemed dangerous, eldery drunks occasionally congregate around the front of Daejeon Korail station at night, including weeknights. Ironically, they are more inclined to hurl abuse at or chase down groups of obvious foreigners than individuals.
Daejeon's tap water is "drinkable" however residents tend to boil their water or purchase affordable bottled water, or fill their own bottles at local parks or at the mountain springs that surround town.
More recently, as of late 2010, the large chaebol megacompanies have started buying up land around the station sites on the proposed second subway line, some of which have since become the scene of local protests and seiges. It is strongly recommended you avoid these areas particularly if the black cars are in the area or you have a visible camera on your person.