It means "currently" from the perspective of the writer.
Surely you're familiar with the English tenses, right? The present tense in the English is equivalent to the present tense in the Greek.
So, John 5:24 means that when one believes, they possess (have) eternal life. That's when it is received.
This link will answer your questions:
http://www.ntgreek.org/learn_nt_greek/inter-tense.htm
The present tenses are equivalent in Greek and English.
I happen to teach NT Greek and some of what you have stated here is incorrect. In English, the tenses primarily relate to the time of action. We add extra words to get kind of action. We could say, 'I go', but to indicate progressive action, we say, 'I am going'.
In Greek (except for the future tense), the tenses refer primarily to the
kind of action.
Therefore, the present tense in Greek is not equivalent to the present tense in English. The Greek present tense refers to continual/continuous action. The time factor is of minor importance.
NT Greek grammarians, Dana & Mantey, stated this important difference when compared with English tenses:
The distinctive function of the verb is to express action. Action as presented in the expression of a verbal idea involves two elements, time of action and kind of action. That is, the action may be described as occurring at a certain time, and must be described, if intelligible, as performed in a certain manner. Tense deals with these two aspects of verbal expression, kind of action being the chief idea involved, for time is but a minor consideration in the Greek tenses.... The important element of tense in Greek is kind of action (Dana & Mantey 1955:177, 178 emphasis in original).
What is the meaning of the present tense in Greek? The aorist tense may be represented by a dot (•). It happened. The present tense by a line (_______________), and the perfect tense by a combination of the two (•_________________) [Dana & Mantey 1955:179].
The fundamental significance of the present tense is the idea of progress. It is the linear tense. This is not, however, its exclusive significance. It is a mistake to suppose "that the durative meaning monopolises the present stem" (M. 119). Since there is no aorist tense for present time, the present tense, as used in the indicative [mood], must do service for both linear and punctiliar action. But it is to be borne in mind that the idea of present time is secondary in force of the tense. The time element belongs to the indicative [mood], where the present tense is really the "imperfect of present time," while what we know as the imperfect tense is the "imperfect of past time." The progressive [i.e. continual/repeated action] force of the present tense should always be considered as primary, especially with reference to the potential moods, which in the nature of the case do not need any "present punctiliar" tense (Dana & Mantey 1955:181, emphasis in original).
We can apply this understanding of the Greek present tense to John 5:24 (ESV): 'Truly, truly, I say to you, whoever hears [present tense] my word and believes [present tense] him who sent me has [present tense] eternal life. He does not come into judgement, but has passed from death to life'.
Therefore the verse means that those who hear Jesus' word and continue to believe him continue to have eternal life. The verse does not teach that a person who once believed and no longer believes has eternal life. Eternal life is for those who continue to believe. That's what the Greek teaches because the Greek present tense is
not equivalent to the English present tense.
John 5:24 is in harmony with Matthew 24:9-14 (ESV),
Then they will deliver you up to tribulation and put you to death, and you will be hated by all nations for my name's sake. 10 And then many will fall away and betray one another and hate one another. 11 And many false prophets will arise and lead many astray. 12 And because lawlessness will be increased, the love of many will grow cold. 13 But the one who endures to the end will be saved. 14 And this gospel of the kingdom will be proclaimed throughout the whole world as a testimony to all nations, and then the end will come (emphasis added) .
Oz
Works consulted
Dana, H E & Mantey, J R 1927/1955,
A Manual Grammar of the Greek New Testament. Toronto, Canada: The Macmillan Company.