In Chinese restaurants, asking 打包 (dǎbāo) is very common and not considered embarrassing at all — it's seen as practical and avoiding waste. The anti-waste campaign (光盘行动 guāngpán xíngdòng) even encourages it.
usage
At a restaurant, just say 打包 and the waiter will bring containers. For packing luggage, 打包行李 is the standard phrase.
