的确 and 还: indeed; 还 as a tone of surprise or admiration
【四12】语气副词 / adverbs
Tone colorsā tone 1á tone 2ǎ tone 3à tone 4a neutral
的确 (“really, indeed”) confirms that something is true, often to agree with someone or to settle a doubt: 这的确是我的错, 他的确没这么说. It goes after the subject, before the verb or adjective, in a statement (a negative is fine: 他的确没来). It asserts; it does not ask.
还 does not mean “still” here. Before 真 or 挺 + verb or adjective, it adds a note of surprise or admiration: 他还真有办法 (he's really resourceful, isn't he!), 这家店的面条还挺好吃的 (the noodles here are quite good, actually). Before a negative it shows impatience: 你怎么还不起床？ (why aren't you up yet?).
Compare 真, which simply intensifies (他真有办法: he is very resourceful), 竟然, which stresses an unexpected fact, and 其实, which corrects an idea. The ironic 还X呢 is a separate point. 的确 is neutral; 还真 and 还挺 are rather spoken.
Examples
这的确是我的错。
Zhè díquè shì wǒ de cuò.This really is my mistake.
他的确没这么说，是你听错了。
Tā díquè méi zhème shuō, shì nǐ tīng cuò le.He really didn't say that; you misheard.
这个办法的确不错，我们试试吧。
Zhège bànfǎ díquè búcuò, wǒmen shìshi ba.This method is indeed good, so let's try it.
他还真有办法，问题马上就解决了。
Tā hái zhēn yǒu bànfǎ, wèntí mǎshàng jiù jiějué le.He's really resourceful: the problem was solved straight away.
这家小店的面条还挺好吃的，下次再来吧。
Zhè jiā xiǎo diàn de miàntiáo hái tǐng hǎochī de, xià cì zài lái ba.The noodles in this little shop are actually pretty good, so let's come again next time.
都十点了，你怎么还不起床？
Dōu shí diǎn le, nǐ zěnme hái bù qǐchuáng?It's already ten o'clock. Why aren't you up yet?
Common mistakes
✗ 他的确来了吗？ → ✓ 他真的来了吗？ (的确 asserts, it is not used to ask a question).
✗ 他还有办法 → ✓ 他还真有办法 (to say “he's really resourceful” you need 真 or 挺: without them, 还 means “still”).