The Hebrew word karov קרוב means "near." All the verbs that derive from the root of that word - קרב - mean "to come near, approach". In Biblical Hebrew, we find that meaning in the kal (karav), piel (kirev), and hifil (hikriv) forms. The hitpael form - hitkarev התקרב - only appears in Hebrew literature after the biblical period.
The form hikriv has an additional meaning. Rabbi Amnon Bazak, in his book Nekudat Peticha (p. 219) points out that for the first two books of the Torah, hikriv means "to approach" (e.g. Bereshit 12:11, Shemot 14:10). However, in the beginning of the book of Vayikra, we find a new meaning:
דַּבֵּר אֶל־בְּנֵי יִשְׂרָאֵל וְאָמַרְתָּ אֲלֵהֶם אָדָם כִּי־יַקְרִיב מִכֶּם קָרְבָּן לַיהוָה מִן־הַבְּהֵמָה מִן־הַבָּקָר וּמִן־הַצֹּאן תַּקְרִיבוּ אֶת־קָרְבַּנְכֶם׃
Speak to the Israelite people, and say to them: When any of you presents an offering of cattle to the LORD, he shall choose his offering from the herd or from the flock. (Vayikra 1:2)
Here hikriv means "bring an offering" and we also find the first mention of the nouns korban קרבן - "offering, sacrifice." Bazak points out that there were many sacrifices earlier in the Torah, but they always use other words like mincha מנחה (Bereshit 4:3), olah עולה (Bereshit 8:20) and zevach זבח (Bereshit 46:1). So why did the Torah start using the word korban only now?
He says that this is due to the meaning of the verb hikriv. Since previously it meant "to draw close to", he claims that only in Vayikra, when God established a permanent location in the Sanctuary, could these sacrifices be considered a way to become near to God. Previously, there might have been a spiritual closeness in sacrifices. Now, when one could actually approach the sanctuary, there was a physical dimension that expressed itself in this new word - korban.
In his book Midabrim Besefat Hatanach, Rubik Rosenthal notes (p. 140-141), that in Modern Hebrew, the words hikriv and korban have left that earlier meaning regarding ritual sacrifices, and split into two different meanings. The verb hikriv means sacrifice in the secular sense: to give up something important for a higher purpose (and the noun form of this verb is hakrava הקרבה - "self-sacrifice."). Korban, however, refers to someone harmed or killed by someone else's action - i.e. a victim. So for example, victims of terrorism are korbanot hateror קרבנות הטרור. There were those that opposed such usage, because the religious sense of korban would seem to instill a higher purpose to those who perpetrated the crimes. But as we've seen many times before, language has a path of its own, and that usage stuck.
A word that derives from this root is krav קרב - "battle." Klein says it probably originally meant "hostile approach." In Israel, a combat soldier is called kravi קרבי - "ready for battle."
A different word that at first glance looks like it should be from the same root, but perhaps isn't is kerev קרב - "midst, interior." The Ben-Yehuda dictionary provides three possibilities:
a) That kerev derives from karav. If this is the case, kerev originally meant something like "drawn close, closeness."
b) They could be from entirely different roots. This what Klein suggests:
1 midst, interior. 2 inward part, bowels, intestines. [Related to Moabite בקרב (= in the midst of), Ugar. qrb (of same meaning), Akka. qirbu (= inward part, interior), qirib (= in). These related words show that the orig. meaning of קֶרֶב was ‘midst, interior’, and that the meaning, ‘inward part, bowel, intestines’ is secondary. However, according to several scholars, Heb. קֶרֶב is related to Arab. qalb (= heart); see קבל ᴵ.His reference to קבל (to be opposite, which we've discussed before) brings us this entry:
BAram. לָקֳבֵל, JAram. קְבֵל, לִקְבֵל (= in front, before), Syr. מֶן קֽבוֹל (= opposite); whence Aram.–Syr., also BAram. קַבֵּל, Heb. קִבֵּל, ‘he received, accepted’), Arab. qabila (= he received, accepted), OSArab. קבל (= to receive, accept), Ethiop. qabala (= he went to meet, encountered), Akka. qablu (= battle; middle of the body, middle). However, according to some scholars Akka. qablu in the meaning ‘middle’ is related to קֽרָב (= battle). According to other scholars Akka. qablu in the meaning ‘middle’ is related to Arab. qalb (= heart; see קֶרֶב)
So according to Klein's approach, the Arabic word qalb - "heart" - developed into two different meanings. One developed into the Hebrew kibel - "to receive, accept" and the other kerev - "interior, inner part." Neither are related to karov - "near."
c) The third possibility mentioned in Ben Yehuda is that one root split into two meanings. This seems to be the approach of Gesenius, whose entry for kerev mentions the Arabic qalb but says that here the "r" softened into an "l" - i.e. the Arabic meaning came later. This understanding would allow that even qalb is related to karov.
